Scattering Theory

Sean Harris

October 2016

A thesis submitted for the degree of Bachelor of Philosophy (Honours) of the Australian National University

Declaration

The work in this thesis is my own except where otherwise stated. The ﬁrst four chapters are largely expository, covering background material. The ﬁnal two chapters contain mostly new theorems (to the best of my knowledge) proven by myself, with some help by others such as Professor Andrew Hassell.

Sean Harris

Acknowledgements

Most of all I would like to thank my supervisor Professor Andrew Hassell, for developing my interest in this topic, always ﬁnding time to meet with me to discuss issues, and being generally great. I am also incredibly grateful to Dr Melissa Tacy for helping me draft this thesis while Andrew was absent. Thanks also goes to the many lecturers I have met leading up to this moment, for nourishing my interest in mathematics. My fellow Dungeon-dwelling honours students have also been incredibly helpful throughout the year. Special mention to Hugh McCarthy, for always being available to chat about any problems I was having (both mathematical and otherwise). A few more people to mention: my tutorial students, for giving me some reprieve from study; the staﬀ at the ANU Food Co-op, for the great lunches (I wish I had discovered them earlier); and the many green envelopes and the single red envelope, for keeping me going. I guess I should thank my parents too.

Abstract

Scattering theory studies the comparison between evolution obeying “free dynamics” and evolution obeying some “perturbed dynamics”. The asymptotic nature of free and perturbed evolution are compared to determine properties of the perturbation. A brief introduction to scattering theory and inverse scattering problems is given in Chapter3, after covering some relevant analysis concepts and the construction of Laplacian and Dirichlet to Neumann operators in Chapters1 and2. The spectral duality result of [EP95] is as follows.

Theorem ([EP95], Main Theorem). The following are equivalent:

2 1. −∆D has an M-fold degenerate eigenvalue k0

2. As k ↑ k0, exactly M eigenphases θj(k) of SD(k) converge to π from below.

In Chapter4 this result is explained both mathematically and intuitively, and the signiﬁcance of the form of the second equivalent statement is examined. Possible generalisations of the main theorem of [EP95] to potential scattering are then investigated in Chapter5. The Cayley transform generalisation, as explained in Section 5.3, suggests studying an object which is given the name transmission operator. The transmission operator displays qualities similar to the scattering matrix and so is investigated further in Chapter6. Many results about the spectrum of the transmission operator are proven under certain physically reasonable conditions on the considered potential for scattering.

vii

Introduction

Our path to Scattering theory starts with the following problem:

Suppose we wish to determine properties of a person’s brain without intrusive testing. One possible way proceed would be to pass waves through the head, and observe the scattered waves which exit. We would like to know that this scattering information can return useful information, and so we will frame this as a mathematical problem. To ﬁnd a starting point for such a theory, we may use the following thought experiment: Suppose n you stand at the edge of a large lake in R (this is mathematics, so the lake will of course be a ball). On top of this, suppose it is 2 a.m. and you are wearing your nicest shoes. In the middle of the lake, you can vaguely make out the shape of an object. Naturally you wish to determine properties of this object but you can’t see it very well due to the time of day, and you don’t want to ruin your shoes by swimming out to the object. All you can do is make splashes on the boundary of the lake, and measure the returning echoes (see Figure1). From this wave scattering information, would it be possible to gain any knowledge of the object?

Figure 1: You stand on the edge of a lake and splash at the mysterious object. The grey solid line is the incoming and reﬂected wave, and the grey dotted line is the path the wave would have taken if there was no object.

ix x

The mathematical framework behind trying to solve such a problem is known as scattering theory. There are a few varieties of scattering theory, and the two we’ll focus on are potential and obstacle scattering, which both fall under the heading of concrete scattering theory. Potential scattering can be thought of as modelling waves passing through localised inhomogeneity in a media (such as a change of refractive index), while obstacle scattering can be thought of as modelling waves travelling through space with some forbidden region (such as electromagnetic waves travelling through a region with a conductive chunk of metal in the way). Such a theory clearly has many real world applications, and it also has many interesting mathematical results! One such result is the spectral duality result of Eckmann and Pillet, given in their paper [EP95]. This spectral duality roughly states that an obstacle D appears “almost invisible” to some incoming radiation at frequencies approaching a given frequency k0, if and only if D can support standing waves of the same frequency k0. The aim of this thesis is to explain the result of Eckmann and Pillet and to attempt to generalise the result to potential scattering. To do this, we shall start by covering some basic analysis facts, using [RS80] and [Are15] as main recource. These will be used to construct the Laplacian - a key component of scattering theory, and the Dirichlet to Neumann operators - a key component of our attempts to generalise the result of [EP95]. The Laplacian is a differential operator which governs the mathematical understanding of many physical processes. To us, its significance will be in passing from physical waves travelling through a lake to mathematical waves travelling through wherever the reader believes mathematics to live. The Dirichlet to Neumann operators relate the boundary values of some function satisfying certain physical laws on a domain, to the flux of said function through the boundary. These operators will be used to come up with a “local problem” to match the potential scattering problem, in the same way the standing wave problem matches the scattering off D. After constructing these operators, we will explain how to pass from our physical problem of mystery objects in lakes and brain scans to a mathematical framework, known as scattering theory (following [Mel95]). This will leave us with enough knowledge to tackle the proof of Eckmann and Pillet’s, of which we will cover the main ideas. Finally, in the last two chapters we shall develop some possible extensions of the spectral duality result to potential scattering. One of these extensions seems quite promising, and a few results concerning it are proven. The promising extension is given in terms of an operator, which we have named the “transmission operator”. The transmission operator shows many similarities to the scattering matrix, a key component of scattering theory and the result of [EP95]. A proposed analogue of the main result of [EP95] related to the transmission operator is given in Section 5.3, as ConjectureB. A slightly modified form of this conjecture is proven for central potentials in Section 6.1, which is a new result of the author’s. Many preliminary results towards a proof of ConjectureB are also proven as new results. All unreferenced results of Chapter6 are the author’s work besides Lemma 6.10, which was joint work with Professor Andrew Hassell. Contents

Acknowledgementsv

Abstract vii

Introduction ix

Notation xiii

1 Background Analysis Concepts1 1.1 Unbounded Operators...... 1 1.2 Diﬀerentiability and Holomorphicity in Banach Spaces...... 5 1.3 The Spectral Theorem...... 6 1.4 Forms...... 10 1.5 Groups and Semigroups...... 11

2 The Laplacian and Dirichlet to Neumann Operators 15 2.1 The Laplacian...... 15 2.1.1 The Laplacian in Spherical Coordinates...... 17 2.2 The Spectral Theorem applied to ∆...... 19 2.3 Dirichlet to Neumann Operators...... 20 2.3.1 Dirichlet to Neumann Operator with respect to −∆...... 21 2.3.2 Dirichlet to Neumann Operator with respect to −∆ + A ...... 22

3 Scattering Theory 25 3.1 Trivial Scattering...... 25 3.2 Non-Trivial Scattering...... 27 3.3 Inverse Scattering Problem...... 30 3.3.1 Potential Scattering Inverse Problem...... 30 3.3.2 Obstacle Scattering Inverse Problems...... 31 3.4 Abstract Scattering Theory...... 32

4 Review of Spectral Duality for Planar Billiards 35 4.1 Spectral Duality - Easy Version...... 35 4.2 Diﬃculties with a Converse...... 36

xi xii CONTENTS

4.3 Spectral Duality - Hard Version...... 37 4.3.1 Potential Theory...... 38

4.3.2 Relating Ak to SD(k)...... 39 4.3.3 Proof of Theorem 4.5...... 41

5 Generalisations of Spectral Duality to Potential Scattering 45 5.1 Dirichlet to Neumann Boundary Condition Generalisation...... 45 5.2 Comparison of Scattering Matrices Generalisation...... 46 5.3 Cayley Transform Generalisation...... 47 5.3.1 Occurrence in Literature, and Notational Conventions...... 50

−1 6 Properties of CV (k)C0(k) 53 6.1 The Central Potential Case...... 56 6.2 Invariance Under Change of Domain...... 58 6.3 Spectral Properties...... 59 6.3.1 Accumulation of Eigenvalues...... 60 6.3.2 Flow of Eigenvalues...... 64 6.3.3 1 as Eigenvalue...... 74

Bibliography 79 Notation

H A separable Hilbert space over R or C.

ONB Orthonormal basis.

(·, ·) An inner product on a given Hilbert space. If multiple Hilbert spaces are being considered, the inner product will be distinguished by subscripts. We use the convention that the inner product is conjugate linear in the second variable: (af, g) = a(f, g).

(·, ·,..., ·) A tuple of elements from a given set.

V A a vector space over R or C.

n |x| The standard Euclidean norm of x ∈ R , or the norm of a number x ∈ C. The distinction will be clear from the context.

||v|| The norm of a vector V in a Banach space. If multiple norms are considered, they will be distinguished by subscripts.

Γ(T ) The graph of the linear operator T .

D(T ) The domain of the linear operator T .

σ(T ) The spectrum of the linear operator T .

ρ(T ) The resolvent set of the linear operator T .

L(X), L(X,Y ) The set of bounded linear transformtations from Banach space X to itself. The set of bounded linear transformations from Banach space X to Banach space Y .

χΩ The characteristic function of the set Ω. I.e. for a set X containing Ω, χΩ : X → R is the function which is identically 1 on Ω, and 0 elsewhere.

∆ The functional Laplacian, with sign convention based on Pn ∂ . i=1 ∂xi ∆ The Dirichlet Laplacian on U ⊂ n, with sign convention based on Pn ∂ . U R i=1 ∂xi

xiii xiv NOTATION

F The Fourier transform.

Λ,C A classical Dirichlet to Neumann operator and its Cayley transform, respectively.

Λ, C A semi-classical Dirichlet to Neumann operator and its Cayley transform, respectively.

SV (k),SD(k) The scattering matrix at frequency k for potential V or obstacle D with Dirich- let boundary conditions, respectively.

1 2 n H (D),H (D) The p = 2 Sobolev spaces on D ⊂ R .

1 ∞ 1 H0 (D) The closure of Cc (D) in H (D).

Df The vector of weak partial derivatives of a function f in some Sobolev space (or for f diﬀerentiable). Chapter 1

Background Analysis Concepts

This chapter will review some basics of the analysis techniques necessary throughout the rest of this thesis. These basics are more advanced than what is typically shown in an undergraduate degree in mathematics, but are still not the main component of this thesis so results will be given without proof. Most of the information below, and the relevant proofs, can be found in either [RS80] or [Are15]. Basic knowledge of Hilbert spaces and bounded operators will be assumed (roughly at an Analysis 3 level). However, the unfamiliar reader may ﬁnd the earlier chapters of [RS80] a good reference for these concepts. Knowledge of Sobolev spaces will also be assumed. See [Eva10] for information. We will use the same notation as in [Eva10], which is repeated in Notation.

1.1 Unbounded Operators

There are many nice properties and definitions concerning bounded operators on Hilbert spaces, such as the notions of self-adjointness and compactness, and the spectral theorem for compact self-adjoint operators. However, in the real world bounded operators are incredibly limited. From a physics and PDE point of view, we would often like to know things about differential operators on function spaces. But differential operators are typically not bounded, as functions may have small norm, but vary wildly. This leads one to consider operators which are not required to be bounded. This section will discuss the main definitions and results for such operators. First recall

Deﬁnition 1.1. A bounded operator on a Hilbert space H is a linear map T : H → H, such that there exists a constant C > 0 with ||T x|| ≤ C||x|| for all x ∈ X. The set of bounded operators on H is written L(H).

This definition is too restrictive for our needs. We would still like some symmetry with respect to the inner product to be satisfied by the operators we consider. According to the Hellinger- Toeplitz Theorem (see Section III.5 of [RS80]), any operator T defined on all of H such that (T φ, ψ) = (φ, T ψ) for all φ, ψ ∈ H must necessarily be bounded. So we suppose that unbounded operators may not necessarily be defined on all of the Hilbert space we are working on, as is the case with differential operators on L2 spaces.

1 2 CHAPTER 1. BACKGROUND ANALYSIS CONCEPTS

Deﬁnition 1.2. An operator A on H is a linear map from the domain of A, a linear subspace D(A) ⊂ H, to H.

Most of the operators we encounter will have dense domain, so this will be assumed unless specified otherwise. Such a general definition does lead to some issues with analysis, in that we know nothing about how the operator interacts with limits. So we make another definition, in an attempt to control limits.

Deﬁnition 1.3. The graph of an operator T is the set

Γ(A) := {hφ, Aφi|φ ∈ D(A)} ⊂ H × H.

The product space H × H can be equipped with the inner product

(hψ1, φ1i, hψ2, φ2i) := (ψ1, ψ2) + (φ1, φ2) making H × H into a Hilbert space. A is called closed if Γ(A) is closed in H × H.

Note that the graph of an operator is a subspace of H × H, and we can always identify an operator with its graph. Depending on the domain which A is deﬁned on, A may not be closed. However, if the domain is adjusted, it might be possible to deﬁne a closed operator extending A.

Deﬁnition 1.4. Let A, A0 be operators on H. If Γ(A) ⊂ Γ(A0), then we say A0 is an extension of A, and write A ⊂ A0. That is, D(A) ⊂ D(A0) and A and A0 agree on D(A).

Deﬁnition 1.5. An operator A is called closable if it has a closed extension. A closable operator A has a smallest closed extension, which is denoted A.

One way to generate closed extensions of an operator A is to take the closure of its graph, Γ(A). The issue with this is that the closure of the graph may no longer be the graph of an operator, that is it may contain points of the form h0, ψi for ψ 6= 0. However, if A is closable, then it is easy to check that Γ(A) = Γ(A).

Deﬁnition 1.6. Let A be a densely deﬁned operator on H. Let D(A∗) be the set of φ ∈ H for which there is an η ∈ H with

(Aψ, φ) = (ψ, η) for all ψ ∈ D(A).

For each φ ∈ D(A∗), we deﬁne T ∗φ = η as above. T ∗ is called the adjoint of T .

It is clear that the adjoint as defined above is an operator, and that if A ⊂ B, then B∗ ⊂ A∗. Note that if A did not have dense domain, then the above definition would not be well-defined (if D(A) is not dense, D(A)⊥ is non-trivial and we could add any element of D(A)⊥ to η while preserving the defining equality as above). In the case that A is a bounded operator, this definition agrees with the usual definition of the adjoint constructed using the Riesz representation theorem. But in the case of unbounded operators, D(A∗) can be quite messy (even possibly just {0}!). If the adjoint A∗ does have dense domain, then we can define the double adjoint A∗∗ := (A∗)∗. In that case, we have the following helpful theorem: 1.1. UNBOUNDED OPERATORS 3

Theorem 1.7 ([RS80], Theorem VIII.1). Let A be a densely deﬁned operator on H. Then:

1. A∗ is closed.

2. A is closable if and only if D(A∗) is dense, in which case A = A∗∗.

3. If A is closable, then (A)∗ = A∗.

This theorem allows us to generally consider closable operators instead of their closures when considering things such as self-adjointness, which is typically very helpful in calculations.

Deﬁnition 1.8. Let A be a closed operator on H. A complex number λ is in the resolvent set of A, denoted ρ(A), if λI − A is a bijection of D(A) onto H with bounded inverse. If λ ∈ ρ(A), −1 Rλ(A) = (λI − A) is called the resolvent of A at λ

Theorem 1.9 (First Resolvent Formula, [RS80], Theorem VI.5). Let A be an operator on H. Then for λ, µ ∈ ρ(A), we have

Rλ(A) − Rµ(A) = (µ − λ)Rµ(A)Rλ(A).

The verification of this fact is a simple computation. There are a few other significant properties of the resolvent, which will be covered in the next section on differentiability and holomorphicity of Banach space valued functions.

Deﬁnition 1.10. Let A be a closed operator on H. The spectrum of A, denoted σ(A), is the compliment of ρ(A) in C. The spectrum is further divided into the point spectrum and the residual spectrum. The point spectrum consists of all those λ for which there exists a non- zero φ ∈ D(A) such that Aφ = λφ, in which case λ is called an eigenvalue of A, with φ the corresponding eigenvector (sometimes called eigenfunction, eigenstate). The residual spectrum is everything in the spectrum that is not an eigenvalue.

We may sometimes refer to the spectrum of a closable, but not necessarily closed, operator, in which case we will always mean the spectrum of the closure. Now we can deﬁne what it means for an operator to be self-adjoint. Note that this is a bit more complicated than in the bounded case, as we may have issues with domains. For this reason, we have an intermediate notion of a symmetric operator

Deﬁnition 1.11. A densely deﬁned operator A on H is called symmetric if A ⊂ A∗. In other words (Aψ, φ) = (ψ, Aφ) for all φ, ψ ∈ D(A).

Deﬁnition 1.12. A is called self adjoint if A = A∗.

Note that by Theorem 1.7, A∗ is closed. So if A is symmetric then A ⊂ A∗, so A∗ is a closed extension of A, so must also extend A = A∗∗. So for symmetric operators we have A ⊂ A∗∗ ⊂ A∗, for closed symmetric operators we have A = A∗∗ ⊂ A∗, and for self adjoint operators we have A = A∗∗ = A∗. Note also that in the case A is bounded, this deﬁnition is equivalent to the usual deﬁnition of self adjointness for bounded operators. 4 CHAPTER 1. BACKGROUND ANALYSIS CONCEPTS

It turns out that this definition of self adjointness for unbounded operators is the correct one as these domain equalities allow one to form functional calculi, which we will see is incredibly important to scattering theory. It should be noted that actually checking an operator is self adjoint directly from the definition could be quite difficult. However, there are many nice theorems which give simple conditions to check self-adjointness, and to construct self adjoint extensions of symmetric operators. While these theorems give a wonderful classification of self adjoint extensions, they are not relevant to the material of this thesis. The interested reader should check [RS75]. As in the case of bounded self adjoint operators, it is easy to check that the spectrum of a self adjoint operator lies completely on the real axis. This will be important in the next section, when we define a functional calculus on self adjoint operators. There are a slew of other definitions concerning operators, which give corresponding results on the spectrum. We say a self adjoint operator A is positive if (Ax, x) ≥ 0 for all x ∈ D(A), or 2 that it is bounded below if there exists an M ∈ R with (Ax, x) ≥ M||x|| . Note that a positive operator is bounded below with M = 0. We will use the notation A ≥ 0 to mean that A is a positive operator, and A ≥ B to mean A − B ≥ 0. Given A bounded below by M (or A ≥ MI), it is quite easy to show that A − λI is invertible for any λ < M, and hence the spectrum of A is bounded below by M. The proof of this is given as an exercise at the end of Chapter 8 of [RS80]. We will combine this knowledge with the following theorem and information about compact operators to obtain one of the most necessary theorems of this thesis, concerning self adjoint operators bounded below, with compact resolvent.

Theorem 1.13 (Hilbert-Schmidt Theorem, [RS80], Theorem VI.16). Let K be a compact self adjoint operator on H. Then there exists an ONB {φn} for H so that Kφn = λnφn for each n, and λn ∈ R. If H is inﬁnite dimensional, then λn → 0 as n → ∞.

Recall also that the set of compact operators, denoted Com(H), form an ideal in L(H), that is Com(H) is a subspace of L(H), and for any S, T ∈ L(H) and K ∈ Com(H), SKT ∈ Com(H). It is also common knowledge (amongst analysts) that Com(H) is norm closed and closed under taking adjoints. These properties are proven in [RS80]. Note that the ﬁrst resolvent formula (Theorem 1.9) combined with these facts shows that if an operator A has compact resolvent Rλ(A) for some λ ∈ ρ(A), then Rµ(A) is compact for all µ ∈ ρ(A). We can thus refer to operators with compact resolvent at some given point in their resolvent set as operators with compact resolvent. These results can be combined to give the following desired theorem, concerning self adjoint operators, bounded below with compact resolvent.

Theorem 1.14. Let A be a self adjoint operator on H bounded below with compact resolvent. Then there exists an ONB {φn} for H consisting of eigenvectors of A with real eigenvalues {λn}. If H is inﬁnite dimensional, then λn → ∞ as n → ∞.

The proof of this is quite simple, so will be outlined. Since A is bounded below, RM (A) exists for some M ∈ R suﬃciently low. Since A has compact resolvent, RM (A) is compact. Using self adjointness of A it can be checked that RM (A) is self adjoint (which requires M ∈ R). Theorem 1.2. DIFFERENTIABILITY AND HOLOMORPHICITY IN BANACH SPACES 5

1.13 is then applied to RM (A) to obtain an ONB {φn} for H consisting of eigenvectors of RM (A) with eigenvalues {µn}. It is then easy to check that each φn is an eigenvalue of A, with eigenvalue λ = M − 1 . The convergence of eigenvalues is then proven by the analogous statement in n µn Theorem 1.13 and using the fact that A is bounded below. As one last result we include the following theorem, attributed to Riesz and Schauder, concerning compact operators not necessarily assumed to be self adjoint. This will play a major role throughout the later chapters of this thesis. Theorem 1.15 (Riesz-Schauder Theorem, [RS80], Theorem VI.15). Let K be a compact operator on H. Then σ(K) is discrete, having no limit points except possibly at 0. Further, any non-zero λ ∈ σ(K) is an eigenvalue of ﬁnite degeneracy.

1.2 Diﬀerentiability and Holomorphicity in Banach Spaces

There are many nuances to coming up with a definition of “differentiable” or “holomorphic” for Banach space valued functions. Differentiating operator valued functions and applying theorems of complex analysis will feature heavily in later chapters. Rather than giving a complete description of the different differentiability conditions, we will now give a very brief overview and some examples. For the most part, everything works as it does for complex valued functions. For a Hilbert space valued function f : U ⊂ R → H, the derivative is defined in the obvious way: f(t + h) − f(t) f 0(t) = lim . h→0 h It is easy to check that this notion of derivative satisfies the usual properties of a derivative, such as linearity and the following nice relationship with the inner product:

(f, g)0 = (f, g0) + (f 0, g).

The only Banach-space valued functions we will encounter will be operator valued. Typically these will have unbounded derivative, so we use the strong deﬁnition of the derivative. We say an operator valued function A on U ⊂ R has derivative A˙(t) at t ∈ U, an unbounded and densely deﬁned operator on H, if for all v ∈ D A˙(t) the following holds:

A(t + h)v − A(t)v lim = A˙(t)v. h→0 h 2 2 Example 1.16. For t ∈ [0, ∞), let A(t): R → L ([0, 1]) be the operator sending the pair 2 (a, b) ∈ R to the solution of the following boundary value problem: ( d2 2 (− dx2 − t )u(x) = 0 on (0, 1) u(0) = a, u(1) = b.

˙ 2 2 Then at t0 ∈ [0, ∞), A(t0): R → L ([0, 1]) sends the pair (a, b) to the solution of: ( d2 2 (− dx2 − t0)u ˙(x) = 2t0u(x) on (0, 1) u˙(0) = 0, u˙(1) = 0. Which is found by diﬀerentiating the equation governing A(t)(a, b) with respect to t. 6 CHAPTER 1. BACKGROUND ANALYSIS CONCEPTS

We will also be confronted with situations in which understanding holomorphicity will be required. For a Hilbert space valued function f from some open set U ⊂ C to H, holomorphicity is defined exactly as one would expect. Namely, for each z ∈ U the following limit is required to exist: f(h + z) − f(z) lim . |h|→0 h Hilbert space valued holomorphic functions enjoy many of the properties of complex valued holomorphic functions, such as local expressions as power series, infinite regularity and integral formulae. It is much more difficult to define a notion of holomorphic operator valued functions (also called a holomorphic family of operators). If we restrict to functions with image in the bounded operators, then we may say such functions are holomorphic if they admit local norm convergent power series representations.

Example 1.17. For an unbounded operator T on H, the resolvent Rλ(T ) is holomorphic. To see this, ﬁx some λ0 ∈ ρ(T ). If we then consider λ close to λ0, we can ﬁnd

∞ ! X n n Rλ0 (T ) I + (λ0 − λ) [Rλ0 (T )] n=1 −1 is a norm convergent power series for |λ − λ0| < ||Rλ0 (T )|| , and is the inverse of (λI − T ). Hence ρ(T ) is open, and Rλ(T ) is a holomorphic operator valued function. As a corollary, the spectrum of any operator is closed. This example was appropriated from Theorem VI.5 of [RS80].

Note that if A(z) is a holomorphic family of bounded operators in L(X,Y ) for Banach spaces X,Y , then for all x ∈ X, y∗ ∈ Y ∗, both A(z)x and y∗(A(z)x) are holomorphic functions (Banach space valued and complex valued respectively). In fact, it can be shown that a family B(z) of bounded operators from X to Y is holomorphic if and only if for all x ∈ X the function B(z)x is holomorphic, if and only if for all y∗ ∈ Y ∗ the function y∗(B(z)x) is holomorphic. This equivalence makes it much easier to check holomorphicity. As mentioned at the beginning of Section 1.1, we will need to consider unbounded operators. The previous deﬁnition of holomorphicity clearly fails for unbounded operator valued functions so we will have to come up with a new one. This is done in [Kat95], Chapter 7. However, the constructions in [Kat95] are complicated and space consuming (over 60 pages!) so will be left to the reader to digest. Fortunately, every unbounded family of operators we will need to consider can be linked to a bounded family of operators via the spectral theorem of the next section, and so we may just bootleg the results from the bounded case.

1.3 The Spectral Theorem

In this section, we will outline the main results known as the Spectral Theorem. There are many different faces of the spectral theorem, and each is very powerful and relevant to the aims of this thesis. The first form of the spectral theorem allows us to “take functions” of a self-adjoint operator A on H. In the case that H is finite dimensional, it is a common result that H has an orthonormal 1.3. THE SPECTRAL THEOREM 7 basis {φi} given by eigenvectors of A, such that each φi has real eigenvalue λi. Then given any function f : {λi} → C, we can define f(A) by setting f(A)φi = f(λi)φi and extending via linearity. We would like to apply this idea to the case that H is infinite dimensional, and the following theorem says that we can.

Theorem 1.18 (Spectral Theorem - Functional Calculus Form, [RS80], Theorem VIII.5). Let A be a self adjoint operator on H. Then there exists a unique map φˆ from B(R), the bounded Borel functions on R, to L(H) so that:

1. φˆ is an algebraic *-homomorphism, i.e.for all f, g ∈ B(R), λ ∈ C,

φˆ(fg) = φˆ(f)φˆ(g) φˆ(λf) = λφˆ(f) φˆ(1) = I φˆ(f) = φˆ(f)∗.

2. ||φˆ(f)|| ≤ ||f||∞.

3. Suppose fn is a sequence of bounded Borel functions on R with hn(x) → x for each x ∈ R and |hn(x)| ≤ |x| for all x and n. Then for all ψ ∈ D(A), φˆ(hn)ψ → Aψ.

4. If hn → h pointwise and ||hn||∞ is bounded, then φˆ(hn) → φˆ(h) strongly.

5. If Aψ = λψ, then φˆ(f)ψ = f(λ)ψ.

We will often write φˆ(f) as f(A) in analogy with the construction in the finite dimensional case, and refer to φˆ as a “functional calculus”. One of the most important applications of this form of the spectral theorem will be discussed in the later section on groups and semigroups. There is an important point which is missed in the above theorem given the level of generality we have approached with. In the construction of the functional calculus for bounded operators, one can start with defining f(A) for continuous functions f, defined only on σ(A). These are then used to extend the functional calculus to include all bounded Borel functions on R. But the functional calculus is still only “supported” on σ(A), in the sense that f(A) = f ◦ χσ(A)(A). This also holds for the unbounded self-adjoint operators A, although in that case this implicit dependence on only σ(A) is hidden even deeper within the proofs. With this in mind, we have the following incredibly important theorem, a generalisation of property (v) in the previous theorem:

Theorem 1.19 (Spectral Mapping Theorem, [RS80], Theorem VII.1(e)). Let A be a self adjoint operator on H, and f be a bounded real valued Borel function on σ(A). Then σ(f ◦ χσ(A)(A)) is equal to the essential range of f, i.e.

{λ ∈ R; ∀ > 0, µ(m ∈ M; |f(m) − λ| < ) > 0}.

In the case f is continuous on σ(A), then σ(f ◦ χσ(A)(A)) = f(σ(A)).

The second guise of the spectral theorem we will consider is incredibly important to scattering theory, and essentially gives rise to the Fourier transform. 8 CHAPTER 1. BACKGROUND ANALYSIS CONCEPTS

Theorem 1.20 (Spectral Theorem - Multiplication Operator Form, [RS80], Theorem VIII.4). Let A be a self adjoint operator on H. Then there exists a measure space hM, µi with µ a ﬁnite measure, 2 a unitary operator U : H → L (M, dµ) and a real valued measurable function f : M → R which is ﬁnite µ-a.e. such that:

1. ψ ∈ D(A) if and only if f(·)(Uψ)(·) ∈ L2(M, dµ).

2. If φ ∈ U(D(A)), then (UAU −1φ)(m) = f(m)φ(m).

Remark 1.21. In applications, we will typically be able to find M and µ explicitly. In some instances, it will be more convenient to drop the finite assumption on µ, as we shall see in Chapter 2. In the context of Theorem 1.20, the image of f and the spectrum of A can be directly related. Suppose for example that f is identically equal to λ ∈ R on a measurable set B ⊂ M with µ(B) 6= 0. Then χB(m) 6= 0 and (f(m) − λ) χB(m) = 0. We find:

−1 −1 −1 A U χB = U (fχB) = λU χB.

−1 So A has eigenvalue λ with eigenvector U χB 6= 0. Now, suppose that λ is in the essential range of f (see Theorem 1.19 for the deﬁnition of essential range). Then for each n ∈ N, choose a measurable set Bn ⊂ M of positive measure such 1 that |f(m)−λ| < n for all m ∈ Bn. Let φn be the characteristic function of Bn, scaled to have unit 2 2 L (M, µ) norm. Then it is easy to check that (f(m) − λ)φn → 0 in L (M, µ). By using continuity −1 −1 −1 of U , it can be seen as before that (A − λI)U φn → 0. But since ||U φn|| = ||φn|| = 1, this implies that (A − λI) is not invertible, and so λ ∈ σ(A).

Before moving on to the last form of the spectral theorem, we need to use the previous theorem to specify some distinguished subsets of σ(A). For a fixed self adjoint operator A, a vector ψ ∈ H is said to be cyclic if {g(A)ψ|g ∈ C∞(R)} is dense in H (where C∞(R) is the set of continuous functions vanishing at infinity). The Riesz-Markov Theorem (see [RS75]) guarantees the existence of a Borel measure µψ on R of finite mass, such that for all g ∈ C∞(R): Z (ψ, g(A)ψ) = g(λ)dµψ(λ). R

The measure µψ is known as a spectral measure. In the case that ψ is cyclic for A, the spectral 2 measure allows us to form a unitary map H → L (R, dµψ), such that A is mapped to multiplication by λ. In general, we can decompose H into a direct sum of (possibly inﬁnitely many) subspaces such that each subspace has a cyclic vector, in which case we obtain a unitary map L∞ 2 H → i=1 L (R, dµψi ). For more information about this construction, see [RS75]. Now, for any ψ ∈ H, we can decompose the spectral measure µψ into a sum as µψ,pp + µψ,ac + µψ,s, where µψ,pp is a pure point measure on R, µψ,ac is absolutely continuous with respect to Lebesgue measure on R, and µψ,s is singular continuous with respect to Lebesgue measure (see [RS75] for deﬁnitions and proof). This then gives the following theorem: 1.3. THE SPECTRAL THEOREM 9

Theorem 1.22 ([RS80], Theorem VII.4). Let A be a self adjoint operator on H. Let Hpp = {ψ ∈

H |µψ is pure point}, Hac = {ψ ∈ H |µψ is absolutely cts} and Hs = {ψ ∈ H |µψ is singular cts}.

Then H = Hpp ⊕ Hac ⊕ Hs, with each subspace invariant under A. A| Hpp has a complete basis of eigenvectors, A| Hpp has only absolutely continuous spectral measures and A| Hs has only singular continuous spectral measures.

From here we split the spectrum into distinguished subsets.

Deﬁnition 1.23.

σpp(A) = {λ|λ is an eigenvalue of A},

σcont(A) = σ(A|Hcont = Hac ⊕ Hs),

σac(A) = σ(A|Hac),

σs(A) = σ(A|Hs).

Our deﬁnition used above is following the conventions used in [RS80]. Some authors use diﬀerent notation, setting σpp(A) = σ(A| Hpp). With our conventions, we have the following theorem:

Theorem 1.24.

σcont(A) = σac(A) ∪ σs(A),

σ(A) = σpp(A) ∪ σcont(A).

This decomposition is incredibly important to scattering theory, which some mathematicians think of as the study of the absolutely continuous spectrum.

The ﬁnal form of the Spectral Theorem which we will consider takes some more work to understand. First note that from the functional calculus form of the spectral theorem, for any measurable set Ω ⊂ R, we can form the operator PΩ = χΩ(A). Then due to the algebraic *-homomorphism properties of the functional calculus, we ﬁnd that:

2 ∗ 1. Each PΩ is an orthogonal projection. That is, PΩ = PΩ = PΩ.

2. P∅ = 0,PR = I. F∞ PN 3. If Ω = n=1 Ωn then PΩ = s-limN→∞ n=1 PΩn .

4. PΩ1 PΩ2 = PΩ1∩Ω2 .

A set of operators satisfying the above properties is called a projection-valued measure

(p.v.m). Note that for any φ ∈ H,(φ, PΩφ) is a well defined Borel measure on R, which is typically denoted by d(φ, Pλφ). We can then use the polarisation identity to define d(ψ, Pλφ) for all ψ, φ ∈ H. For a measurable Borel function g : R → C, we can define g(A) by setting Z 2 Dg = {φ| |g(λ| d(φ, Pλφ) < ∞}, R 10 CHAPTER 1. BACKGROUND ANALYSIS CONCEPTS and letting g(A) act on Dg via Z (φ, g(A)φ) = g(λ)d(φ, Pλφ), R using polarization to obtain g(A). We can write this symbolically as Z g(A) = g(λ)dPλ. R

It is quite easy to check that Dg is dense in H, and that if g is real valued then g(A) with domain R Dg is self adjoint. Note that we have that A = λdP if {P } is the family of spectral projections R λ Ω related to A. This leads to the ﬁnal form of the Spectral Theorem which we will consider.

Theorem 1.25 (Spectral Theorem - Projection Valued Measure Form, [RS80], Theorem VIII.6). There is a one to one correspondence between self adjoint operators A and projection-valued measures {PΩ} on H given by Z A = λdPλ. R If g is a real valued Borel function on R, then Z g(A) = g(λ)dPλ, R deﬁned on Dg is self adjoint. If g is bounded, g(A) agrees with φˆ(g) of Theorem 1.18.

1.4 Forms

Working directly with operators can be quite diﬃcult, and most operators occurring in physics tend to be obtained from something known as a form. Working with forms has many beneﬁts over working with operators, making them typically the better choice for applications. We only need concern ourselves with bounded forms for the sake of this thesis, so will follow the material given in [Are15].

Deﬁnition 1.26. A sesquilinear form (or simply a form) on a vector space V over R (or C) is a map a : V × V → R, (C), such that a is linear in the ﬁrst variable and conjugate-linear in the second variable.

There are a few helpful properties of forms which we will need to cover. The form a is called symmetric if for all u, v ∈ V a(u, v) = a(v, u), and accretive if

If a form is accretive and symmetric, it will be called positive. We will also use the notation a(u) := a(u, u) for the quadratic form associated to a. 1.5. GROUPS AND SEMIGROUPS 11

If V is in fact a Hilbert space, we have some further deﬁnitions to give. We say the form a is bounded if there exists some M ≥ 0 such that for all u, v ∈ V we have

|a(u, v)| ≤ M||u||V ||v||V and coercive if there exists some α > 0 such that

With these deﬁnitions in hand, we can start to construct operators from forms.

Theorem 1.27 ([Are15], Propositions 5.5, 6.15, and Theorem 6.10). Let V, H be Hilbert spaces over R (C) and let a : V × V → R(C) be a bounded form. Let j ∈ L(V,H) be an operator with dense range. Suppose further that if u ∈ V with j(u) = 0 and a(u) = 0, then u = 0. Then

A = {(x, y) ∈ H × H; ∃u ∈ V : j(u) = x, a(u, v) = (y, j(v))∀v ∈ V } deﬁnes an operator on H, called the operator associated to the pair (a, j) and denoted by A ∼ (a, j). Furthermore, if j is compact then A has compact resolvent, if a is symmetric then A is self adjoint, and if a is positive then A is positive also.

This is quite a new way of obtaining operators from forms, and is quite powerful. There are many other useful properties of A which are difficult to check, but can be determined directly from (much more easily verified) properties of a and the previous theorem. These are typically used for generating semigroups, and the interested reader may refer to [Are15] for more information. Typically, j is an embedding of V into H, but particularly interesting operators can be obtained when V and H are very different spaces. We will see examples of both instances while constructing the Dirichlet Laplacian and Dirichlet to Neumann operators in Chapter2. Note that if a is coercive, then the requirement in the preamble of Theorem 1.27 is guaranteed. Alternatively, if there exists some ω ∈ R, α > 0 such that

2 2

1.5 Groups and Semigroups

Below is one form of the heat equation in one dimension, t as the time variable and x as the spatial variable ∂u ∂2 ∂t = ∂x2 u for all t > 0 u(0, x) = g(x) for all x ∈ R, ∂2 with initial condition g in some function space on R. If we pretend that the operator A = ∂x2 is a constant, then we could form the symbolic solution

u(x, t) = etAg(x). 12 CHAPTER 1. BACKGROUND ANALYSIS CONCEPTS

The basic motivation of semigroup theory is to make sense of the above expression, and use it to understand the dynamics of evolution equations.

Deﬁnition 1.28. Let X be a real or complex Banach space. A one-parameter semigroup on X is a function T : [0, ∞) → L(X) satisfying

T (s + t) = T (s)T (t), for all t, s ≥ 0.

If additionally lim t(t)x = x, for all x ∈ X, t→0 then T is called a C0-semigroup or a strongly continuous semigroup. If T is deﬁned on R and the ﬁrst property holds, T is a one-parameter group, and if the second condition also holds

T is called a C0-group

Example 1.29. The simplest example of a C0-group is the usual exponential. Given some ﬁxed at a ∈ R, we deﬁne T (t) on the Banach space R by T (t)x = e x. It is easy to check that T is a C0-group.

Example 1.30. As another simple example, let X be any Banach space, and fix some A ∈ L(X). Define T (t) by ∞ X (tA)j T (t) := etA = . j! j=0 It is easy to check that the sum above converges in norm (by using the triangle inequality, the fact that ||AB|| ≤ ||A|| ∗ ||B|| for all A, B ∈ L(X), and the convergence of the power series for the exponential on R). It is also quite easy to check that the defining properties of a C0-group are satisfied by T (t). This example almost solves our issue with the heat equation mentioned at the ∂2 beginning of this section, except that ∂x2 is unbounded (depending on its domain of definition)! In the previous example, we say that etA is the semigroup generated by A. It turns out that all C0-semigroups have generators, in the following sense.

Deﬁnition 1.31. For a C0-semigroup T , we say A is the generator of T , where A is the linear operator with graph T (h)x − x A = {(x, y) ∈ X × X|y = lim exists}. h→0+ h

There are some important properties of the generator of a C0-semigroup which we need to know before being able to study them. These are collected below:

Theorem 1.32 ([Are15], Theorem 1.10). Let T be a C0-semigroup on X, with generator A. Then the following hold:

1. For x ∈ D(A), T (t)x ∈ D(A) for all t ≥ 0, then function t 7→ T (t)x is continuously diﬀeren- tiable on [0, ∞), and d T (t)x = AT (t)x = T (t)Ax. dt 1.5. GROUPS AND SEMIGROUPS 13

R t 2. For all x ∈ X, t > 0 one has 0 T (s)xds ∈ D(A), and Z t A T (s)xds = T (t)x − x. 0

3. D(A) is dense in X, and A is a closed operator.

4. Let t0 > 0, and let u : [0, t0) → X be continuous, u(t) ∈ D(A) for all t ∈ (0, t0), u continuously 0 diﬀerentiable on (0, t0), and u (t) = Au(t) for all t ∈ (0, t0). Then u(t) = T (t)u(0) for all

t ∈ [0, t0).

5. Let S be a C0-semigroup on X, with generator B ⊃ A. Then S = T , B = A.

It is the fourth property above that really relates semigroups to the solution of evolution equations. Of course, we are much more interested in starting with an operator A, and then ﬁnding a

C0-semigroup so that A is its generator, as this will then allow us to solve the evolution equation:

d dt u(t) = Au(t) for all t > 0 u(0) = g.

There is a large theory relating spectral properties of a given operator A to when A can be used to generate a semigroup, and what properties the resulting semigroup achieves. However, only a very small part of this theory is relevant to the topics of this thesis. For a more elaborate view of semigroup generation, see [Are15].

What we will focus on is the generation of C0-semigroups by self-adjoint operators on a Hilbert space H. In this case, we actually want to construct the semigroup eitA, which will provide solutions to the equation: d −i dt u(t) = Au(t) u(0) = g. This is then related to the wave equation:

d2 2 dt2 u(t) = −A u(t) u(0) = g.

So eitA is often considered a “wave operator”, where the above equation is a generalisation of the d2 ∂2 wave equation dt2 u(x, t) = ∂x2 u(x, t). This is the basis of the “abstract” scattering theory, set forth at the end of Chapter3. Since we are only interested in self adjoint operators, we can apply the spectral theorem in its functional calculus form (Theorem 1.18) to deﬁne eitA for A self-adjoint. This immediately implies the following properties:

Theorem 1.33. Let A be a self adjoint operator on H and deﬁne U(t) = eitA. Then:

1. For each t ∈ R, U(t) is a unitary operator (that is, (U(t)φ, U(t)ψ) = (φ, ψ) for all φ, ψ ∈ H, ∗ −1 or equivalently U(t) = U(t) ). Also, for all s, t ∈ R, U(s + t) = U(s)U(t).

2. For all φ ∈ H and t → t0, U(t)φ → U(t0)φ. 14 CHAPTER 1. BACKGROUND ANALYSIS CONCEPTS

U(h)φ−φ 3. For φ ∈ D(A), h → iAφ as h → 0.

U(h)φ−phi 4. If limh→0 h exists, then φ ∈ D(A).

We use U because the group is unitary, rather than T , which is more commonly associated with translation. A family of unitary operators indexed by R satisfying properties 1 and 2 above is called a C0- or strongly continuous one-parameter unitary group. The following theorem of Stone’s says that all strongly continuous one-parameter unitary groups are obtained by exponentiating iA for some self adjoint operator A.

Theorem 1.34 (Stone’s Theorem, [RS80], Theorem VIII.8). Let U(t) be a strongly continuous one- parameter unitary group on H. Then, there is a self-adjoint operator A on H such that U(t) = eitA.

The importance of unitarity should be made clear. The basic dynamics of quantum mechanics follow an equation of the form d −i ϕ(t) = Hϕ(t), dt in which H is the Hamiltonian, a self adjoint operator on a “state space”. Observable quantities in quantum mechanics are given by other self adjoint operators, A, in which an observation is formed by ﬁnding (ϕ(t), Aϕ(t)). Under certain commutation conditions on A and H (which I will not spell out, as commutativity is understandably complicated for unbounded operators, due to domain considerations), we can commute A and U(t) to ﬁnd

(ϕ(t), Aϕ(t)) = (U(t)ϕ(0), AU(t)ϕ(0)) = (U(t)ϕ(0),U(t)Aϕ(0)) = (ϕ(0), Aϕ(0)).

So we obtain measurements which do not change in time from self adjoint operators which commute with the Hamiltonian. Chapter 2

The Laplacian and Dirichlet to Neumann Operators

In this section, we will construct two of the most important operators in scattering theory, the Laplacian and the Dirichlet to Neumann operators, and determine some of their properties. We n will (almost) only consider these operators on open subsets of R , but the deﬁnitions and theorems can be extended quite naturally to smooth manifolds.

2.1 The Laplacian

The Laplacian is a basic component of many PDE governing physical situations, such as the diffusion of heat and the propagation of waves. n The standard Laplacian on an open set U ⊂ R is defined to be the operator ∆ acting on a function f ∈ C2(U) as n X ∂2 ∆f := f. ∂x2 i=1 i We will take this sign convention. In this case −∆ will be a positive operator. This definition suffices for C2(U), and has many nice integration properties and relations to the Fourier transform n 2 (if U = R ). But C (U) is a difficult space to work over, especially when it comes to showing existence and uniqueness of solutions to PDE. So we need to come up with ways of extending the Laplacian to act on better spaces, ideally L2(U). The good news is that this can be done, by using a the machinery already covered Chapter1. To motivate our extensions, recall the following theorem:

n Theorem 2.1 (Green’s Formula, [Eva10], Theorem 3 p.628). Let U ⊂ R be open with smooth 2 boundary, u, v ∈ C (U). Let dn and dS be the outward facing normal derivative operator and surface measure on ∂U respectively. Then:

R R R 1. U Du · Dvdx = − U v∆udx + ∂U vdnudS R R 2. U (v∆u − u∆v)dx = ∂U (vdnu − udnv)dS

15 16 CHAPTER 2. LAPLACIAN AND DTN

If we take v compactly supported in U in the ﬁrst formula, then Z Z Du · Dvdx = − v∆udx. U U

The left hand side of this looks like a form on H1(U), while the right hand side is an inner product of v with −∆u (up to taking the complex conjugate of v). This suggests that we might be able to use our form methods to deﬁne a reasonable extension of the Laplacian to L2(U). Instead, we will 2 start by extending to Lloc(U). This brings us to our ﬁrst extension of the Laplacian.

2 Definition 2.2. The functional Laplacian ∆ is defined as the operator on Lloc(U) with domain ( ) u ∈ H1 (U); ∃f ∈ L2 (U) s.t. (Du, Dv) = (f, v), D(∆) = loc loc ∀v ∈ H1(U) compactly supported defined by ∆u = −f.

This is the functional definition of the Laplacian, which only acts “locally”. It is easy to check 2 2 Pn ∂2 that Hloc(U) ⊂ D(∆), and for u ∈ Hloc(U), ∆u = i=1 2 u in the weak sense. The domain of the ∂xi 2 Laplacian defined above is actually larger than Hloc(U) for certain U, although we will see later n 2 n 2 n that if U = R , then D(∆) ∩ L (R ) = H (R ) exactly. Definition 2.2 will suffice for defining Dirichlet to Neumann operators later in this chapter, but this definition does not have many nice properties beyond that. For the sake of Scattering Theory, we wish to be able to specify some kind of boundary conditions on solutions of PDE governed by a Laplacian-style operator. This is achieved by modifying the domain of the form considered above, and using our constructions relating forms and operators given in Section 1.4.

We will run through the construction of the Dirichlet Laplacian, ∆U , which has domain

1 2 D(∆U ) = {u ∈ H0 (U); ∆u ∈ L (U)}. (2.1)

The Dirichlet Laplacian can be used to model the diffusion of heat or the propagation of waves when the dynamics are required to vanish on the boundary. Again, it is interesting to note that if the boundary of D is not nice enough then it is possible to find elements of D(∆U ) which are not 2 2 1 in H (U), although H (U) ∩ H0 (U) ⊂ D(∆U ) always, with the same action as in the local case. 1 1 To construct ∆U , we will consider the classical Dirichlet form, a : H0 (U) × H0 (U) → C defined by Z a(u, v) = Du · Dvdx. U Note that a is bounded, as

|a(u, v)| ≤ ||Du||L2(U)||Dv||L2(U) ≤ ||u||H1(U)||v||H1(U).

1 2 Now, if we let j : H0 (U) → L (U) be the usual inclusion, we ﬁnd that

2 2 |a(u)| + ||j(u)||L2(U) = ||u||H1(U). 2.1. THE LAPLACIAN 17

So a is j-elliptic. Hence we can form the operator A associated to (a, j), using Theorem 1.27. We claim (and it is easy to verify), that A = −∆U as deﬁned in Equation 2.1. For a more detailed explanation of this construction, see Example 5.13 of [Are15]. Note −∆U is positive and self adjoint by Theorem 1.27, since a is positive and symmetric, and so has spectrum lying in [0, ∞). In the case that U is bounded with continuous boundary, the Rellich-Kondrachov Theorem

(Theorem 7.11 of [Are15]) states that j is compact, in which case Theorem 1.27 states that −∆U has 2 compact resolvent. So L (U) has an ONB consisting of eigenvectors of −∆U with real eigenvalues, due to Theorem 1.14, and these eigenvalues must be real, discrete and converging to +∞. This will be an incredibly important property when we consider Concrete Scattering Theory in Chapter3. ∞ A similar construction can be used to show that for V ∈ L (U) real valued, the operator −∆U +V is self adjoint, bounded below, with compact resolvent and so L2(U) has an ONB consisting of eigenvectors of −∆U +V with real eigenvalues, which are discrete and converge to +∞ (see [Are15] for details).

n For an open set U ⊂ R , we call the equation

(−∆ − k2)u = 0 on U (2.2) the Helmholtz equation at frequency k, or energy k2, on U. Note that this uses the functional n Laplacian, so there are no assumed boundary conditions. For example, if U = R , we obtain for each n−1 ikx·θ θ ∈ S a solution Φ0(k, x, θ) = e of the Helmholtz equation at frequency k. These solutions do not lie in any reasonable Lp space besides L∞. There are many local regularity theorems concerning solutions of the Helmholtz equation, although as this example shows global regularity is not guaranteed. In this way, the Helmholtz equation doesn’t necessarily give eigenfunctions of the Laplacian on a sensible space. However, these solutions are incredibly important to scattering theory, as we shall see in Chapter3.

2.1.1 The Laplacian in Spherical Coordinates

Throughout some of the calculations performed in later chapters we will wish to use some spherical symmetry arguments, decomposing functions into a certain collection of functions called spherical harmonics. This will allow us to perform calculations on the spherical harmonic decomposition, in which symmetry arguments are best suited. For this purpose, we will introduce the spherical Laplacian. It will become apparent that this operator will only be applied to suﬃciently smooth functions, and so can just be given in terms of an explicit formula rather than as an unbounded operator with speciﬁed domain. 2 n Given a function f(x) ∈ C (U) on some open set U ⊂ R with continuous boundary, we may x n−1 pass to spherical coordinates r = |x| ∈ [0, ∞), θ = |x| ∈ S , in which case we may consider f as n−1 a function from some subset of [0, ∞) × S to C given by f(r, θ) = f(rθ), whenever rθ ∈ U. The standard Laplacian on U can then be expressed in these new coordinates, with the formula:

∂2 n − 1 ∂ 1 ∆f(r, θ) = f(r, θ) + f(r, θ) + ∆ f(r, θ), (2.3) ∂r2 r ∂r r2 tan 18 CHAPTER 2. LAPLACIAN AND DTN

2 n−1 n−1 where ∆tan is the tangential Laplacian acting on C (S ), and S is given its usual smooth n structure induced by its inclusion into R . For information about this, see any text on diﬀerential 1 geometry, such as [Can13]. In the case n = 2, the tangential Laplacian on S is given by ∆tang(θ) = ∂2 ∂θ2 g(θ), and in higher dimensions there is a more complicated formula which is still known explicitly. As in our extension of the Laplacian to L2, the tangential Laplacian on Sn−1 may be extended to an unbounded self adjoint operator on L2(Sn−1). In this case the self adjoint extension is unique, because there are no boundary conditions which can be changed (since Sn−1 has no boundary). For a construction of this extension, see Section 7 of [Can13]. We will need some properties of this self adjoint extension:

Theorem 2.3 (Sturm-Liouville Decomposition, [Can13], Theorem 44). There exists an ONB 2 n−1 {Θi(θ)} of L (S ) consisting of eigenfunctions of ∆tan, with eigenvalues {Ci}, such that:

0 ≤ C1 ≤ C2 ≤ ...

∞ n−1 Furthermore, Θi ∈ C (S ) for each i.

The functions Θi are referred to as spherical harmonics. This theorem holds in more generality, replacing Sn−1 with any compact smooth Riemannian manifold. The proof given in [Can13] is very nice, and works by considering diﬀusion under a heat equation. In keeping with our sign n convention for the Laplacian on R , the tangential Laplacian on the sphere has non-positive eigenvalues. For this reason, some consider the Laplacian to be −∆tan. Now we may ﬁnally get to the main purpose of this section, the decomposition of a function into spherical harmonics.

n 2 Theorem 2.4. Given U ⊂ R open bounded and symmetric about the origin, and some f ∈ L (U), there exists functions Ri from an open bounded subset of [0, ∞) to C, such that:

∞ X f(r, θ) = Ri(r)Θi(θ), i=1

2 in the L (U) sense. If f is smooth, each Ri is smooth.

Proof. To begin the proof, recall the fact that given two measure spaces (M, µ) and (M 0, µ0), and 2 2 2 0 ONBs {φi(x)} for L (M, µ), {ψi(y)} for L (M, µ), then {φi(x)ψj(y)} is an ONB for L (M ×M , µ⊗ µ0). This is proven on page 51 of [RS80]. Since U is spherically symmetric around the origin, we may write U = I × Sn−1 for some open bounded set I ⊂ [0, ∞), where r ∈ I and θ ∈ Sn−1. There is some issue with this identiﬁcation if 0 ∈ I. However, note that for the measure on I × Sn−1 to be the Lebesgue measure (up to identiﬁcation of f(r, θ) = f(rθ)), the measure on I must be rn−1dr (this is common knowledge, 2 and can be found in any vector calculus textbook). Then {0} ⊂ R has measure zero, as does {0} × Sn−1 ⊂ I × Sn−1, so both subsets can be ignored in considering L2 spaces. 2 n−1 We then choose some ONB {ψi(r)} for L (I, r dr), in which case the aforementioned theorem 2 combined with the Sturm-Liouville decomposition shows that {ψj(r)Θi(θ)} is an ONB for L (U). 2.2. THE SPECTRAL THEOREM APPLIED TO ∆ 19

Hence we can represent:

∞ ∞ X X f(r, θ) = aijψj(r)Θi(θ) = Ri(r)Θi(θ), i,j=1 i=1

P∞ 2 where Ri(r) = j=1 aijψj(r). Since the sum representing f(r, θ) converges in L and {ψj(r)Θi(θ)} P∞ 2 2 n−1 is an ONB, the sum i,j=1 |aij| must converge, and hence Ri ∈ L (I, r dr). Note by orthonor- mality of {Θi(θ)}, we ﬁnd

Ri(r) = (f(r, θ), Θi(θ))L2(Sn−1) . If f is smooth f will be smooth in the r variable, and hence we can diﬀerentiate the above in r (passing through the integration in the inner product, since everything is well behaved), to ﬁnd that Ri(r) is smooth.

The main use of this theorem is the justification of separation of variables, used as a method of solving PDE with spherical symmetry. We will see many examples of this in Chapter6. This actually requires more work, as we would like to differentiate term by term. In general, if a series of functions converges in L2 the such term-wise differentiation is not valid, so we need improved our method of convergence. Ideally we would like uniform convergence, at least locally. That uniform convergence holds is quite difficult to verify, and will generally require specifying the type of PDE the given function must satisfy, and using regularity and uniqueness arguments. Luckily for us, this has been checked in [Gus99] and will hold in each case investigated in this thesis.

2.2 The Spectral Theorem applied to ∆

Throughout this section we will work with ∆Rn , which is self-adjoint as shown in Section 2.1. So we can use the spectral theorem in the multiplication operator form, Theorem 1.20 applied to ∆Rn . 2 n 2 This provides a measure space (M, µ), a unitary operator U : L (R ) → L (M, dµ) and a real valued function f : M → R which is ﬁnite µ-a.e. such that:

2 1. u ∈ D(∆Rn ) if and only if f(·)(Uu)(·) ∈ L (M, dµ).

−1 2. If φ ∈ U(D(∆Rn )), then (U∆Rn U φ)(m) = f(m)φ(m)

n In this case, we can give (M, µ) and U explicitly. In fact, we can take (M, µ) to be R with Lebesgue measure and U to be the Fourier transform.

Theorem 2.5 ([RS75], Theorem XI.6 and unnumbered Lemma on page 2). The Fourier transform 2 n 2 n F : L (R ) → L (R ) given by Z 1 −ix·ξ F(u)(ξ) = n f(x)e dx (2π) 2 Rn

∞ n for f ∈ Cc (R ) and extended via density, is unitary and satisﬁes the conditions on U in Theorem 2 1.20 with A = ∆Rn and f(ξ) = −|ξ| . 20 CHAPTER 2. LAPLACIAN AND DTN

Since f(ξ) = −|ξ|2 is not constant on any sets of positive measure, the remark following Theorem

1.20 implies that ∆Rn has no eigenvalues. In fact, since f is continuous ∆Rn has only purely absolutely continuous spectrum, equal to Ran f = [0, ∞). Note that the properties given by Theorem 2.5 work for the functional Laplacian ∆ (although the functional Laplacian is not self adjoint) as for ∆Rn whenever the Fourier transform of something in D(∆) can be deﬁned. We also have the familiar result:

1 n Theorem 2.6 ([RS75], unnumbered Lemma, page 2). For u ∈ H (R ) and j ∈ {1, . . . , n} ∂ F(−i f)(ξ) = ξjF(f)(ξ). ∂xj

These properties of the Fourier transform are incredibly useful for solving PDE and analysing systems. For our purposes, the Fourier transform is very important in Concrete Scattering Theory, as covered in Chapter3. We will now prove a useful result using the Fourier transform. Note that it is clear from the n 2 n deﬁnition of the functional and Dirichlet Laplacians on R that D(∆) ∩ L (R ) = D(∆Rn ). So as promised after Deﬁnition 2.2, we shall prove the following theorem:

2 n Theorem 2.7. D(∆Rn ) = H (R )

2 n Proof. Showing that H (R ) ⊂ D(∆Rn ) is trivial. So suppose that u ∈ D(∆Rn ). By Theorems 2.5 2 2 n and 1.20, |ξ| (Fu)(ξ) ∈ L (R ). But then for each i, j = 1, . . . , n:

2 2 n |ξiξj(Fu)(ξ)| ≤ |ξ| |(Fu)(ξ)| ∈ L (R )

2 n So ξiξj(Fu)(ξ) ∈ L (R ). Taking the inverse Fourier transform and applying Theorem 2.6 shows ∂ ∂ u(x) ∈ L2( n). Since this holds for all i, j = 1, . . . , n, we ﬁnd u ∈ H2( n) as claimed. ∂xi ∂xj R R

2.3 Dirichlet to Neumann Operators

Dirichlet to Neumann type operators are incredibly important in applications, so this section will start by motivating their study. n Suppose you have some open set U in R with some density function ρ on U. Think of U as your own head, and ρ as mass density function. Say that you would like to know the value of ρ to determine properties of your head, such as possible medical issues. Ideally you would like to determine properties of ρ without any intrusive measurements of U, so only measurements on ∂U can be made. Suppose we can apply a charge distribution f on ∂U, and then measure the output current Λf from ∂U for each charge distribution f (in L2(∂U), say). The mapping f → Λf is known as a Dirichlet to Neumann operator. This name is used because Λ maps boundary data (Dirichlet conditions) to boundary derivative data (Neumann conditions). Due to the theory of electromagnetism, applying a charge distribution f to the boundary of U will lead to a charge distribution u on U, satisfying a PDE of the form: 2.3. DIRICHLET TO NEUMANN OPERATORS 21

( Lu = 0 on U (2.4) u ≡ f on ∂U, for some second order differential operator L depending on ρ. So, up to some rescaling, Λf = dnu, where f, u are as above, and dn is the outward facing normal. Would knowing Λ be enough to reconstruct ρ? This clearly has many practical purposes, from medical imaging to determining conductivity of a medium to geology. The problem was first posed by Albert Calderón(see [Cal80]). We will consider a special case of this, in which L = −∆ + A, for some real-valued bounded measurable function A on U. This has significant applications in attempting to generalise the results of [EP95] to potential scattering with potential A, as we shall see in Chapter5. In some sense, the Dirichlet to Neumann operator produces reasonable “inside problems” for potential scattering, in analogue to the inside problem of the main result of [EP95], the Dirichlet eigenvalue problem. Before showing that a Dirichlet to Neumann style operator exists for such L we need to generalise the definition of normal derivatives, since solutions of Equation 2.4 with f not smooth will fail to be smooth. The main aim is that an analogue of Green’s Formula (Theorem 2.1) holds. So we have the following definition:

n 1 Deﬁnition 2.8 (Weak Normal Derivative, [Are15], Section 7.3). Let U ⊂ R be bounded with C boundary. Let u ∈ H1(U) with ∆u ∈ L2(U). We say u has (outward pointing) weak normal 2 2 1 derivative dnu ∈ L (∂U) if there exists an h ∈ L (∂U such that for all v ∈ H (U) Z Z Z Du · Dvdx + v∆u = vhdS, U U ∂U

In which case we write dnu = h. Note that if u ∈ C2(U) then the above deﬁnition agrees with the standard notion of normal derivative. So by using the density of C2(U) in H1(U), it can be shown that weak normal derivatives are unique, when they exist. Interestingly, we can deﬁne the class of functions with “dnu = 0” for U open with arbitrarily complicated boundary, as the set of u ∈ H1(U) with ∆u ∈ L2(U) such that for all v ∈ H1(U) Z Z Du · Dvdx + v∆u = 0. U U In this sense, the weak normal derivative is much nicer than the standard normal derivative of second year calculus.

2.3.1 Dirichlet to Neumann Operator with respect to −∆

Now we will construct the Dirichlet to Neumann operator Λ0 for L = −∆ on an open bounded set n 1 U ⊂ R with C boundary, and determine some of its properties. This will again use the machinery of quadratic forms explained in Chapter1, but will use the full force of Theorem 1.27 with j not just an inclusion. The operator we are aiming for has graph

2 2 1 {(f, g) ∈ L (∂U) × L (∂U); ∃u ∈ H (U), −∆u = 0, u|∂U = f, dnu = g}. 22 CHAPTER 2. LAPLACIAN AND DTN

Before we get to constructing such an operator, recall the following:

Theorem 2.9 ([Are15], a conglomeration of many theorems and remarks from Chapter 7). Let n 1 U ⊂ R be open and bounded with C boundary. There exists a unique bounded operator tr : 1 2 H (U) → L (∂U) known as the trace operator, such that tr(u) = u|∂U for all smooth u. The trace operator is compact.

We will typically write tr(u) = u|∂U . We will also extend the classical Dirichlet form Z a(u, v) = Du · Dvdx U 1 1 to the domain H (U) × H (U). We will construct Λ0 as the operator associated with (a, tr) using Theorem 1.27, after the following Lemma.

Lemma 2.10 ([Are15], Proposition 8.1). The form a as above is tr-elliptic.

Now we come to our existence theorem:

Theorem 2.11 ([Are15], Theorem 8.4). The operator A on L2(∂U) associated to (a, tr) via The- orem 1.27 agrees with the desired Dirichlet to Neumann operator Λ0; that is

2 2 1 Γ(A) = {(f, g) ∈ L (∂U) × L (∂U); ∃u ∈ H (U), −∆u = 0, u|∂U = f, dnu = g}.

The operator Λ0 is self-adjoint and positive, with compact resolvent.

Remark 2.12. Applying Theorem 1.14 shows that L2(∂U) has an ONB consisting of eigenvectors of Λ0, each having positive eigenvalue and such that the eigenvalues converge to +∞.

2.3.2 Dirichlet to Neumann Operator with respect to −∆ + A

While the Dirichlet to Neumann operator with respect to −∆ is quite interesting, it is far from good enough for the purposes of the rest of this thesis. To be able to generalise the main result of [EP95] to potential scattering, we need to come up with an “inside problem” for potential scattering. For this, we would like to consider ΛA, the Dirichlet to Neumann operator with respect to the operator L = −∆ + A, where A ∈ L∞(U) and A only takes real values. Such an L can be used for modelling the motion of waves or diﬀusion of heat through an inhomogeneous medium. One issue in this case is that uniqueness of solutions to ( Lu = 0 on U u ≡ f on ∂U are no longer guaranteed (as they are in the case L = −∆). For example, if

0 ∈ σ(−∆U + A),

0 then we can add any u ∈ ker(−∆U + A) to u as above to ﬁnd ( L(u + u0) = 0 on U u + u0 ≡ f on ∂U, 2.3. DIRICHLET TO NEUMANN OPERATORS 23

0 1 0 since u ∈ H0 (U). But then we might ﬁnd that dnu 6= dn(u + u ), so the Dirichlet to Neumann operator can’t be deﬁned uniquely (although it will still be a linear relation). Luckily we have the following theorem, courtesy of [Are15].

Theorem 2.13 ([Are15], Theorem 8.14). Suppose 0 ∈/ σ(−∆U + A). Then

2 2 1 {(f, g) ∈ L (∂U) × L (∂U); ∃u ∈ H (U), −∆u + Au = 0, u|∂U = f, dnu = g} is the graph of a a self-adjoint operator ΛA with compact resolvent. Also, ΛA is bounded below.

The proof of this theorem requires a lot of supplementary results, so will not be given. The interested reader may ﬁnd the proof in [Are15] however.

Remark 2.14. As in the case with A = 0, applying Theorem 1.14 shows that L2(∂U) has an ONB consisting of eigenvectors of ΛA, such that the eigenvalues converge to +∞. 24 CHAPTER 2. LAPLACIAN AND DTN Chapter 3

Scattering Theory

In this chapter we will introduce Scattering Theory in two main ﬂavours: concrete and abstract. The focus will be on concrete scattering theory, as this is most relevant to our objectives. However, the abstract formulation has some important qualities, so will be brieﬂy mentioned at the end. We will start by setting up the mathematical framework behind the real world problem given in the Introduction, which will be referred to as Concrete Scattering Theory. As a basic case, we will start by looking at trivial scattering in which there is no object.

3.1 Trivial Scattering

Given we wish to examine asymptotics of waves, we will start by examining solutions to the wave equation: ∂2 − ∆u(x, t) = − u(x, t), on n × . (3.1) ∂t2 R R

As discussed in Section 2.2, the spectrum of −∆Rn is positive and is purely absolutely continuous. 2 2 n So −∆ has no eigenfunctions in L (noting D(∆) ∩ L (R ) = D(∆Rn )). However, we may consider n generalized eigenfunctions of −∆ on R , that is, bounded solutions of

−∆u(x) = k2u(x),

ikt where we take k real and greater than 0, since −∆Rn has positive spectrum. Then e u(x) will be a bounded solution of Equation 3.1. We exclude the case k = 0, as this corresponds to no evolution in time. So instead of the time dependent equation 3.1, we will look at the generalised eigenfunction time independent equation, which is

2 n (−∆ − k )u(x) = 0, on R , (3.2) for k > 0. This is the Helmholtz equation, as described in Section 2.1. As our physical problem requires examining asymptotics of solutions of the Helmholtz equation, we also impose the radiation conditions: ∂ + ik2 u(rθ) ∈ L2( n). (3.3) ∂r R

25 26 CHAPTER 3. SCATTERING THEORY

x where r = |x|, θ = |x| are the spherical coordinates. We will call solutions of any of the PDE that we encounter which satisfy these radiation conditions scattering solutions. ikx·θ n−1 The keen reader might have noticed that the plane waves Φ0(k, x, θ) := e for θ ∈ S satisfy these two conditions. Essentially, the radiation condition is required to remove the possibility of the plane waves e−ikx·θ, which will be used for uniqueness results later. We can use this to obtain ∞ n−1 a wide class of solutions in the following way. Given g ∈ C (S ), we define the function H0(k)g on n by: R Z (H0(k)g)(x) = Φ0(k, x, ω)g(ω)dω. Sn−1 Checking that H0(k)g satisfies Equations 3.2 and 3.3 and that it is bounded is trivial, by commuting integration and differentiation (valid since everything is smooth and integration is over a compact set) and using the fact that Φ0(k, x, θ) satisfy the same equations.

Now we may apply the Principle of Stationary Phase to obtain an asymptotic form for H0g, x expressed in spherical coordinates r = |r|, θ = |x| :

ikr e − 1 π(n−1)i H0(k)g)(rθ) = e 4 g(θ) 1 (n−1) (kr) 2 −ikr e 1 π(n−1)i 0 − 1 (n+1) + e 4 g (θ) + O(r 2 ), as r → ∞ 1 (n−1) (kr) 2 The Principle of Stationary Phase essentially says that since we are integrating against a highly oscillatory function Φ0(k, x, ω), the main contributions to the integral come from the points where the phase x · ω is stationary in ω, which is at ω = ±θ. See [Ros03] for more information. A consequence of applying this is that

g0(θ) = in−1g(−θ)

To interpret this asymptotic form in the terms of our original problem, we can reintroduce the time dependence, obtaining the travelling wave:

ik(t+r) ik(t−r) e − 1 π(n−1)i e 1 π(n−1)i n−1 − 1 (n+1) e 4 g(θ) + e 4 i g(−θ) + O(r 2 ) (3.4) 1 (n−1) 1 (n−1) (kr) 2 (kr) 2 The ﬁrst term will maintain constant phase eik(t+r) along curves of with r +t, so as t increases r must decrease, representing an incoming wave. Similarly, the second term will have constant phase ik(t−r) − 1 (n−1) e along curves of constant t − r, and so represents an outgoing wave. The factors r 2 cause dissipative nature as r increases. This form provides one lemma, which we require before we can relate this solution to our scattering problem. This lemma provides us with uniqueness of the association g 7→ g0.

Lemma 3.1 ([Mel95], Lemma 1.2). For each k > 0 and g ∈ C∞(Sn−1) there exists a unique scattering solution u of Equations 3.2 of the form

ikr −ikr e e 0 − 1 (n+1) u = g(θ) + g (θ) + O(r 2 ), as r → ∞. 1 (n−1) 1 (n−1) r 2 r 2 Where necessarily g0(θ) = in−1g(−θ) 3.2. NON-TRIVIAL SCATTERING 27

For the proof, see [Mel95]. Now we can relate to our original problem.

Deﬁnition 3.2. The Absolute Scattering Matrix for trivial scattering is the map A0(k): ∞ n−1 ∞ n−1 n−1 C (S ) → C (S ) for k > 0 deﬁned by (A0(k)g)(θ) = i g(−θ).

The terminology “scattering matrix” is a remnant from the one-dimensional scattering theory in which this really is a 2×2 matrix, though in higher dimensions the scattering matrix is an operator on inﬁnite dimensional spaces. It follows directly from linearity of Equation 3.2 and the deﬁnition of

A0(k) that A0(k) is a linear map. It is also clear by deﬁnition that A0(k) is surjective and preserves 2 2 n−1 2 n−1 L norms, so extends to a unitary map A0(k): L (S ) → L (S ). We would like trivial scattering to give trivial scattering matrix, so we normalise to obtain the (relative) scattering 2 n−1 2 n−1 −(n−1) matrix S0(k): L (S ) → L (S ) deﬁned by (S0(k)g)(θ) = i (A0(k)g)(−θ) = g(θ). We will typically refer to the relative scattering matrix by the simpler name “scattering matrix”. The input of the scattering matrix will often be referred to as incoming radiation data, and the output as the corresponding outgoing radiation data or scattering data. Now that we have trivial scattering, we may move on to the problem with some perturbation.

3.2 Non-Trivial Scattering

We will start by specifying our (time independent) classical scattering problems. These come in two ﬂavours. ∞ n There is Potential Scattering, which for potential function V ∈ Cc (R ), is concerned with scattering solutions of the equation:

2 n (−∆ + V − k )u = 0, on R . (3.5)

This can be thought of as a localised inhomogeneity in the refractive index of the material, such that waves can still pass through but do not travel freely. The regularity and decay conditions on V can be weakened, although those chosen above will play an incredibly important role in Chapters 5 and6. n Alternatively, there is Obstacle Scattering which, for D ⊂ R open and bounded with nice enough boundary, looks at scattering solutions of

2 n (−∆ − k )u = 0 on R \D, u|∂D = 0. (3.6)

The boundary conditions we are considering are Dirichlet, although these could be changed. We will require ∂D to be smooth, although this can be signiﬁcantly weakened. This can be thought of as modelling electromagnetic waves travelling through free space containing a conductive chunk D, so the potential inside D must vanish. We will use the terminology obstacle to refer to such a bounded open set D.

Now, to define scattering for these non-trivial cases our plan of attack is as follows. We will first obtain some perturbed plane wave scattering solutions of a similar form to Φ0(k, x, θ). We will use these to build scattering solutions as in the free case. After examining uniqueness of such scattering 28 CHAPTER 3. SCATTERING THEORY solutions, we will the be able to define an absolute and relative scattering matrix as in the free case. This will provide us with enough structure in our mathematical framework to investigate the physical scattering problem. Fortunately, this work has been done for us. We will state the required theorems and give some justification to their proofs, although the proofs themselves are very technical and shall be left out.

Lemma 3.3 ([Mel95], Lemma 2.4). For each k > 0 and θ ∈ Sn−1, there is a unique scattering solution ΦV (k, x, θ) to 3.5 and a unique scattering solution ΦB(k, x, θ) to 3.6 (i.e. both satisfy the radiation condition 3.3) of the form

ikx·θ −ik|x| − 1 (n−1) ΦV,B(k, x, θ) = e + exp |x| 2 φV,B(k, x, θ),

∞ n with φV (k, x, θ), φB(k, x, θ) ∈ C (R ), such that derivatives of all orders exist and have ﬁnite limits as |x| → ∞.

The proof of this lemma is the main focus of a whole chapter in [Mel95], and requires a lot of work with resolvents of operators. These are initially not deﬁned on the real axis, and must be extended continuously to the real axis to obtain solutions of the above form. Uniqueness is another issue entirely, which is shown via considering solutions of the relevant PDE up to quickly decaying smooth error terms. Overall, it is a lot of work so shall not be included here, but the interested reader is urged to read the proof in Chapter 2 of [Mel95]. ∞ n−1 ∞ n As in the trivial case, we can form the operators HV (k),HB(k): C (S ) → C (R ) deﬁned by Z (HV,B(k)g)(x) = ΦV,B(k, x, ω)g(ω)dω. (3.7) Sn−1

The operators HV,B maps to scattering solutions of Equations 3.5 and 3.6 respectively, which can be checked easily using the form of ΦV,B(k, x, θ). We may again apply the Principle of Stationary

Phase to HV,B(k)g and simplify to obtain the next Lemma.

Lemma 3.4 ([Mel95], p.23). For each k > 0 and g ∈ C∞(Sn−1) there exists unique scattering solutions uV , uB of Equations 3.5 and 3.5 respectively of the form

ikr −ikr e e 0 − 1 (n+1) uV,B = g(θ) + g (θ) + O(r 2 ), as r → ∞, 1 (n−1) 1 (n−1) V,B r 2 r 2 where necessarily Z 0 n−1 1 π(n−1)i − 1 (n−1) n−1 gV,B(θ) = i g(−θ) + e 4 (2π) 2 k φV,B(k, θ, ω)g(ω)dω Sn−1

0 Since solutions of this form are unique and the association g 7→ gV,B is linear by linearity of the Equations 3.5 and 3.6, we can make the following deﬁnition as in the trivial case.

Definition 3.5. The absolute scattering matrix for either potential V or obstacle B, AV,B(k): ∞ n−1 ∞ n−1 0 C (S ) → C (S ), is defined to be the linear map g 7→ gV,B of Lemma 3.4. −(n−1) The (relative) scattering matrix is defined by (SV,B(k)g)(θ) = i (AV,Bg)(−θ). 3.2. NON-TRIVIAL SCATTERING 29

As before, we will often refer to the relative scattering matrix by the simpler name scattering matrix. At the moment, our scattering matrix is only deﬁned on C∞(Sn−1). This makes things diﬃcult to analyse, as C∞(Sn−1) is not a particularly nice space. To remedy this we have the following theorem:

Lemma 3.6. Suppose uV is a scattering solution of 3.5 and uV has the asymptotic form

ikr − 1 (n−1) −ikr − 1 (n−1) 0 − 1 (n+1) uV = e r 2 g(θ) + e r 2 gV (θ) + O(r 2 ), as r → ∞

∞ n−1 2 0 for g ∈ C (S ). Then the L norms of g and gV agree.

The proof of this lemma is a basic application of Green’s formula (Theorem 2.1) and the radiation conditions Equation 3.3. An identical lemma with similar proof holds in the case of obstacle scattering. This lemma allows us to extend the scattering matrix SV,B(k) continuously to an operator on L2(Sn−1), via density of C∞(Sn−1) in L2(Sn−1). We will not change notation for this extension. There are some very important basic properties of the scattering matrix, which we will collect below.

Theorem 3.7. For either potential or obstacle scattering, for any k0 > 0 the scattering matrix satisﬁes:

1. S(k0) = I + K where K is a compact operator given by integration against a smooth kernel.

2. S(k0) is unitary.

1 3. The spectrum of S(k0) consists only of eigenvalues, lying on the unit circle S ⊂ C. The eigenvalues accumulate at 1 only, and eigenfunctions for eigenvalue not 1 are smooth.

0 Proof. 1. This follows directly from the form of gV in 3.4. That the operator K is compact is a consequence of its integral kernel being smooth (see [RS80] Chapter VI.5).

2. Note that Lemma 3.6 shows that the extension the scattering matrix to L2(Sn−1) is norm

preserving, so it suﬃces to check that it is invertible. Since S(k0) is a compact perturbation of

the identity by part (1), the Fredholm alternative says that S(k0) is invertible if and only if it

is injective (see [RS80] for details on the Fredholm alternative). But S(k0) is norm preserving

and hence injective, and hence invertible. So S(k0) is unitary.

3. Since S(k0) is I + K for K compact, the spectrum of S(k0) is exactly {λ + 1; λ ∈ σ(K)}. Following the discussion in Section 1.1, the spectrum of compact operators consists only of

eigenvalues, and these eigenvalues accumulate at 0 only. So the spectrum of S(k0) consists only of eigenvalues with the same eigenfunctions as K, which accumulate at 1 only. For an eigenfunction g ∈ L2(Sn−1) with eigenvalue λ 6= 1, we ﬁnd:

1 g = Kg 1 − λ But K has smooth kernel, so the right hand side is smooth and hence g is smooth. 30 CHAPTER 3. SCATTERING THEORY

2 n−1 To see that the spectrum of S(k0) lies on the unit circle, we can take f ∈ L (S ) with eigenvalue λ and ||f|| = 1. Then:

2 2 2 |λ| = ||S(k0)f|| = ||f|| = 1

Unitarity of the scattering matrix is an incredibly important property, and makes sense on physical grounds. Comparing to the time dependent form in Equation 3.4 and its following physical interpretation, the input g of the scattering matrix can be thought of as the incoming waves set oﬀ from the shore of the lake at t = −∞, and the output S(k0)g can be thought of as the outgoing waves returning to the shore at t = ∞ (up to some unitary modiﬁcations incurred by passing from absolute to relative scattering matrix). Then from energy conservation, we would expect the energy contained in the incoming waves to be equal to that of the outgoing waves. This is exactly 2 preservation of L norms. That S(k0) should be invertible comes from expecting that running time backwards should swap incoming and outgoing waves.

Remark 3.8. What does the accumulation of eigenvalues at 1 correspond to in our physical problem? If eigenvalue 1 is obtained, then by comparing Deﬁnitions 3.2 and 3.5 we see that there exists scattering solutions of the free problem and the perturbed problem with the same asymptotic form. In other words, the perturbation makes no impact on scattering, and the perturbation is “invisible” to the given incoming radiation data. The accumulation of eigenvalues at 1 can then be interpreted as meaning that all perturbations on the free problem are “almost invisible” for wild enough incoming radiation data.

With this mathematical formalism in hand, we may return to our initial physical problem.

3.3 Inverse Scattering Problem

We wish to determine properties of either potential V or obstacle B from knowledge of only the scattering matrix SV (k) or SB(k). The good thing is that this can be done! This has not been the main focus of the research undertaken for this thesis, so we shall just state results and give a sketch of how these results are proven, along with commentary on the physical validity of the theorems.

3.3.1 Potential Scattering Inverse Problem

Theorem 3.9 ([Mel95], Corollary 3.1). The map V 7→ SV from (complex-valued) potentials V ∈ ∞ n Cc (R ) to scattering matrix, is injective.

The proof of this theorem requires analysing k → ∞ asymptotics of the smooth kernel of

SV (k) − I. It can be shown that this integral kernel has an asymptotic expansion in k as k → ∞, and the coeﬃcients of the expansion are determined from V via its X-ray transform, which essentially measures averages of V along rays from the origin. The X-ray transform is invertible, and so V can be recovered from the asymptotic expansion. 3.3. INVERSE SCATTERING PROBLEM 31

Some beneﬁts of the proof of this theorem include that it gives a way of actually reconstructing the potential from the scattering matrix. But the proof requires high frequency knowledge of the scattering matrix, which corresponds in our physical problems to either causing tsunamis on the shore of a lake or blasting somebodies head with high energy radiation, both of which aren’t physically feasible. A proof of Theorem 3.9 can be found in [Mel95], in which it is labelled Corollary 3.1. As an alternative, we have the following:

Theorem 3.10 ([Mel95], Proposition 3.5). For any n ≥ 3 and any k > 0, the map from real ∞ potentials V ∈ Cc to scattering matrix at frequency k, SV (k), is injective.

Before explaining the idea behind the proof of this theorem, we shall examine its pros and cons in comparison to the previous theorem. The dimensional requirement is fine in our physical problems, since the universe has spatial dimension at least 3. The reduction to real potentials is also fine for the case of most measurements in our universe. That scattering at only one frequency determines the potential is a clear benefit over the last theorem in which knowledge for asymptotic frequencies was necessary. However, this theorem has the (major) downside that it is not re-constructive: it does not produce a method of determining the potential from the scattering matrix. The proof ∞ of this theorem starts by assuming that two real valued potentials V1,V2 ∈ Cc have the same scattering matrix. It can then be shown that for any g ∈ C∞(Sn−1), the following holds: Z

0 = (V1(x) − V2(x))(HV1 g)(x)(HV2 g)(x) Rn n where HV1,2 are as in Equation 3.7. It is then shown that in dimension n ≥ 3, for any ξ ∈ R , a ix·ξ g can be chosen so that (HV1 g)(x)(HV2 g)(x) approximates e to arbitrary precision. Then the above equality with these chosen g implies that in the limit the Fourier transforms of V1 and V2 2 n agree. Since both V1 and V2 are compactly supported they are both in L (R ), in which case we may invert the Fourier transform to ﬁnd V1 = V2 almost everywhere. Smoothness of V1 and V2 implies equality everywhere. This provides no technique for reconstruction of V . For a complete proof see [Mel95], in which this theorem is labelled Proposition 3.5.

3.3.2 Obstacle Scattering Inverse Problems

As in the potential scattering case, we ﬁrst have a theorem about injectivity of the mapping from obstacle to scattering matrix

Theorem 3.11 ([Mel95], Proposition 5.2). Suppose n ≥ 3 is odd and B1,B2 are two obstacles with ∞ n−1 smooth boundary. Then if there exists f ∈ C (S ) not identically zero such that SB1 (k)f =

SB2 (k)f for all k > 0, then B1 = B2.

As in the second inverse scattering theorem given for potentials, this theorem does not give a method of reconstructing the obstacle from the scattering. It also requires more stringent constraints on the dimension of the space we work over. The parity constraint is incurred during the proof of existence of the perturbed plane wave solutions in Lemma 3.3, in which the resolvent of 32 CHAPTER 3. SCATTERING THEORY

−∆ − k2 must be continuously extended to real k. This was a detail skipped earlier in this Rn\B thesis as it is quite complicated and not very informative. For more information, see the Chapter 2 of [Mel95]. However, this condition is satisfied by the (classical) universe we live in, so does not produce an issue with most applications of this mathematical framework to the real world. As an advantage over the potential scattering inverse problem theorems, this theorem only requires checking one incoming wave, the f ∈ C∞(Sn−1) mentioned in the requirements. This is nice for real world applications, as your favourite scanning machine need only produce one waveform. However, the condition that SB1 (k)f = SB2 (k)f must hold for all k > 0 is a significant issue as in the potential case: physically this corresponds again to either causing tidal waves at the side of a lake or bombarding a person’s head with high-energy radiation. There has been significant work put into reconstructive inverse obstacle scattering problems, due to their applicability to the real world. It was shown by David Colton in [Col84] that for general data there may be no obstacle producing this data as scattering data, and even if the data does correspond to an obstacle, this correspondence is in no way continuous. This lack of continuity makes the typical methods of analysis impossible to use, as approximation requires continuity. For this reason, it seems that most work towards inverse obstacle scattering has been focused on numeric approximation, given some prior assumed knowledge of the obstacle. For example, in [Has11] a method is used in which it is assumed that an obstacle D lies within some chosen and nice bounded open set G, with conditions on G concerning the spectrum of −∆G. The waves scattered off D with Dirichlet boundary conditions can be approximated outside G. These approximations are then approximated further in G\D, and then the boundary of D is found as the zero level set of this further approximation. The methods of approximation have some serious issues with non-convexity in regions of the boundary of the obstacle D. There is another method considered in [Has11] in which a certain modified form of the Poisson kernel which blows up on the boundary of the obstacle is investigated, which can be partially reconstructed from the scattering data. Both of these methods seem to give good approximations of the obstacle shape at least at a qualitative level, based on the diagrams shown in the same paper.

3.4 Abstract Scattering Theory

As promised, we shall now brieﬂy develop the abstract theory of scattering in Hilbert Spaces, following [RS79]. The idea behind abstract scattering theory is to compare the asymptotics of dynamics generated by diﬀerent self-adjoint operators, e−itA and e−itB, one considered as the “free” dynamics (B), with the other a “perturbation” (A). This has obvious similarities to concrete scattering theory, although everything is more abstract (hence the name). We want to compare the evolution of states as t → ∞ under the perturbed dynamics, for states which appear “free” for t → −∞. By this, we mean that there exists a φ+ such that

−itB −itA lim ||e φ+ − e φ|| = 0. t→−∞ Note by unitarity this is equivalent to

itA −itB lim ||e e φ+ − φ|| = 0, t→−∞ 3.4. ABSTRACT SCATTERING THEORY 33 which is just the requirement that a certain strong limit exists. There are some clear issues if φ+ is an eigenvector of B, in that the above strong limit can only exist if φ+ is also an eigenvector of A with the same eigenvalue. For this reason, we make the following deﬁnition:

Deﬁnition 3.12. Let A and B be self-adjoint operators on a Hilbert space, and let Pac(B) be the projection onto the absolutely continuous spectrum of B. We say the generalized wave operators Ω±(A, B) exist if the strong limits

± itA −itB Ω (A, B) = s-lim e e Pac(B) t→∓∞ exist. When Ω±(A, B) do exist, we deﬁne:

± H± = Ran Ω (A, B).

We may drop the A and B from the notation, when the operators in question are apparent from context. It turns out that the strong limit is the best one to use in this deﬁnition. It can be shown that if Pac(B) = 1 then the norm limit exists if and only if A = B. Also, if A has purely discrete spectrum, the weak limit will exist and be zero, regardless of how diﬀerent A and B are. Some important properties of the generalized wave operators are compiled below

Proposition 3.13 ([RS79], Chapter XI.3 Proposition 3). Suppose Ω±(A, B) exist. Then:

± 1. Ω are partial isometries with initial subspace Ran Pac(B) and ﬁnal subspaces H±.

2. H± are invariant subspaces for A, and

Ω±(D(B)) ⊂ D(A),AΩ±(A, B) = Ω±(A, B)B.

3. H± ⊂ Ran Pac(A).

The proof of this proposition is quite trivial, so will be left to the reader. Now to consider scattering, we want to take states which appear asymptotically free as t → −∞ (i.e. those in H+) and map them to evolution under the perturbed dynamics as t → ∞. To do this, we need to take into consideration the range of Ω±. This warrants the following deﬁnition.

Deﬁnition 3.14. Suppose Ω±(A, B) exist. We say they are complete if and only if

H+ = H− = Ran Pac(A).

There are many theorems about sufficient conditions on A and B so that Ω±(A, B) will exist and be complete. These are quite technical and will generally be satisfied for all the operators considered in this thesis. However, the interested reader may find them in [RS79]. Now we can finally define the Scattering operator.

Deﬁnition 3.15. Suppose Ω±(A, B) exist. We deﬁne the Scattering Operator to be

S = (Ω−)∗Ω+. 34 CHAPTER 3. SCATTERING THEORY

Unravelling this definition, we find that S maps a state asymptotically free in the distant past, which evolves under the perturbed dynamics, to a state asymptotically free in the distant future. This exactly matches with the intuition behind the action of the scattering matrix S(k) of concrete scattering theory. ± In the case that Pac(B) = 1, then Proposition 3.13 shows that completeness of Ω (A, B) is equivalent to unitarity of S. Now, we will give some more argument as to how this scattering operator S is related to a scattering matrix S(k), which in some sense only acts at a certain frequency k. It can be seen by the definition of S and Proposition 3.13 part 2 that SB = BS, i.e. the scattering operator commutes with the free dynamics. This implies that it is possible to express S in terms of the spectral resolution of B, as in Theorem 1.25. But for the attentive reader, this is exactly what was done at the beginning of Section 3.1. Rather than looking at scattering with arbitrary frequency components, we restricted to (−∆ − k2)u = 0. (3.8)

Now, there is an issue incurred here in trying to relate the rest of Section 3.1 to the abstract case, in that for the concrete case we could consider generalised eigenfunctions, which actually solve 3.8 and 2 n live in some space containing H = L (R ). Since in the abstract case the deﬁnition of S requires projection onto the absolutely continuous spectrum of B it is unhelpful to consider eigenfunctions. There are a few ways around this, which we shall investigate heuristically. The ﬁrst is the auxiliary space method, in which we embed some Banach space X into H, so that we obtain the natural embedding X ⊂ H ⊂ X∗. Then generalised eigenfunctions are searched for in the larger space X∗. This is explained in [RS79]. The second method uses the multiplication operator form of the Spectral Theorem (Theorem 2 1.20). For this discussion, we will use U for the unitary map H → L (M, µ) and f : M → R as the multiplying map from Theorem 1.20, and we will assume M is a smooth manifold and that f is −1 smooth. Then λ ∈ σ(B) if and only if λ ∈ Ran f. Then for any λ ∈ R with f (λ) a submanifold, the decomposition of the scattering operator S can be thought of as S acting on L2(f −1(λ)), with the induced measure. The space f −1(λ) is referred to as an energy shell. This almost relates identically to what we have done for concrete scattering theory. In concrete n scattering theory B = −∆, U has been the Fourier transform (as discussed in Section 2.2), M = R with Lebesgue measure, and f has been f(ξ) = |ξ|2. The energy shells are then f −1(k2) = {ξ ∈ n 2 2 2 n−1 R ; |ξ| = k }, which is an (n − 1)-sphere of radius k. Rescaling by k, we obtain S(k): L (S ) → L2(Sn−1), the concrete scattering matrix! Chapter 4

Review of Spectral Duality for Planar Billiards

In this section, we will review the results of Eckmann and Pillet given in the paper [EP95]. In brief, the main result of [EP95] relates the spectrum of the scattering matrix SD(k) for an obstacle 2 D ⊂ R with certain regularity conditions and Dirichlet boundary conditions to the eigenvalues of the Dirichlet Laplacian on D itself. At an intuitive level, the result states the obstacle “looks invisible” from a distance at a frequency k0 > 0 (that is, SD(k0) has eigenvalue 1 following Remark

3.8) if and only if the obstacle can support standing waves of frequency k0. However this is a bit simplistic, the true main result of [EP95] is below:

2 1. The Dirichlet Laplacian −∆D on D has an M-fold degenerate eigenvalue k0

2. As k ↑ k0, exactly M eigenvalues of SD(k) converge to 1 in a clockwise direction (recalling 1 that the spectrum of SD(k) only consists of eigenvalues situated on the unit circle S ⊂ C, from Theorem 3.7).

The second statement is not quite equivalent to the obstacle “looking invisible” at frequency k0, being more closely related to the obstacle looking “almost” invisible at frequencies approaching k0 from below. Before getting into an overview of why the second term of this theorem has such a strange form, we’ll see a simpler form of this theorem with a much simpler proof. Here we shall specify some notation for this chapter alone. D will always be a simply connected 2 2 open bounded subset of R , such that ∂D is piecewise C , and that the angle between smooth 2 pieces is bounded away from 0 and 2π. −∆ will be the functional Laplacian on R , while −∆D will be the Dirichlet Laplacian on D (see Section 2.1 for details). Both k and k0 will be used for real positive numbers, typically with k ↑ k0.

4.1 Spectral Duality - Easy Version

As promised, we’ll start by stating and proving a simple form of the spectral duality result of [EP95].

35 36 CHAPTER 4. SPECTRAL DUALITY

2 Theorem 4.1 ([EP95], Theorem 2.2). If SD(k0) has eigenvalue 1 of degeneracy M, then k0 is an eigenvalue of −∆D of degeneracy at least M. Furthermore, the eigenfunctions of −∆D with 2 2 eigenvalue 1 constructed can be extended to bounded solutions of (−∆ − k0)u = 0 on R .

2 1 Proof. Let 0 6= g ∈ L (S ) be an eigenfunction of SD(k0) with eigenvalue 1. By Deﬁnition 3.5 and 2 2 Lemma 3.4, there exists a bounded solution of (−∆ − k0)uD = 0 on R \D with uD = 0 on ∂D of the following asymptotic form:

ikr − 1 −ikr − 1 − 3 uD = e r 2 g(θ) + e r 2 igD(−θ) + O(r 2 ), as r → ∞,

1 1 πi where gD = g for eigenvalue 1 to be obtained. Now, let u0 = k 2 e 4 H0(k0)g as in Section 3.1. 2 2 Then u0 is a bounded solution to (−∆ − k0)u0 = 0 on R and has the asymptotic form:

ikr − 1 −ikr − 1 − 3 u0 = e r 2 g(θ) + e r 2 ig(−θ) + O(r 2 ), as r → ∞.

2 2 2 Let w = uD − u0 on R \D. Then (−∆ − k0)w = 0 on R \D, and w has the asymptotic form:

− 3 w = O(r 2 ), as r → ∞.

2 2 It has been shown in Section 2 of [GRH12] that non-trivial solutions of (−∆ − k0)w = 0 on R \D cannot decay as quickly as w, so w = 0. Hence uD and u0 agree identically outside of D. So u0|D 2 vanishes on ∂D, satisﬁes (−∆ − k0)u0|D = 0. To see that u0|D is not identically 0, we use Theorem 2 3.2 of [CC06] which states that solutions of (−∆ − k0)u = 0 are real analytic, and hence if u0 vanished on D, u0 would have to vanish identically. But this contradicts the fact that g 6= 0. So 2 u0|D is an eigenfunction of −∆D with eigenvalue k0, and can be extended (by u0) to a bounded 2 2 solution of (−∆ − k0)u0 = 0 on R . If the degeneracy of 1 as an eigenvalue of SD(k0) is M then we can repeat this process with diﬀerent g forming a basis for the eigenspace of SD(k0), obtaining

M linearly independent eigenfunctions of −∆D with eigenvalue 1.

It was no exaggeration that this theorem is an easy version of the main result of [EP95].

Remark 4.2. The proof of Theorem 4.1 provides existence of two scattering solutions u0, uD to the free problem and the problem with obstacle respectively, such that u0 and uD have the same asymptotic form. This argument can be reversed, to show that if such u0, uD exist, then SD(k0) has eigenvalue 1. This equivalence of eigenvalue 1 with existence of free and perturbed scattering solutions can be proven for potential scattering in a similar way, to show that the scattering matrix

SV (k0) has eigenvalue 1 if and only if there exist scattering solutions u0, uV of the free and perturbed problems at frequency k0 respectively, which agree identically outside of the support of V .

4.2 Diﬃculties with a Converse

It would be nice if the converse of Theorem 4.1 would hold. There are many issues with this though, which we will now investigate.

The ﬁrst issue relates to the method of proof of Theorem 4.1. The function u0 was shown to be real analytic and also to vanish on ∂D, which shows that ∂D is an analytic set (by deﬁnition). 4.3. SPECTRAL DUALITY - HARD VERSION 37

But there are plenty of domains D which satisfy our regularity conditions, but have boundaries that are not analytic sets. Another similar issue in reversing the argument used in the proof of

Theorem 4.1 is that of extending eigenfunctions of −∆D outside of D to obtain eigenfunctions of

SD(k0). Here we can give an explicit and simple example of where this fails:

Example 4.3 (The irrational cake, [EP95], Example 1a). Let D be the circular wedge of radius one and angle rπ, for r ∈ [0, 2] irrational. The eigenfunctions of −∆D can be computed explicitly by separating variables in polar coordinates. The radial component will be a rπ periodic trigonometric function. But since r is irrational, a continuation of the angular function will not possibly be 2π 2 periodic, so the eigenfunctions cannot be extended to all of R .

So, given that a direct converse of Theorem 4.1 cannot possibly hold, maybe we could instead consider eigenvalues as k → k0 without any monotonicity assumption for the moment. Then we are hit with another issue. As was shown in Theorem 3.7, 1 is an accumulation point of the eigenvalues of SD (in fact it is the only accumulation point). But it is not generally possible to track eigenvalues through an accumulation, as is shown in Section 3 of [GRH12]. So it may not even make sense to consider the limit as k → k0. The good thing is that accumulation only occurs on one side:

Theorem 4.4 ([EP95], Lemma 2.1). For each k > 0, the eigenvalues of SD(k) accumulate at 1 only from below. I.e. only ﬁnitely many eigenvalues have positive imaginary part.

So the only way we could be sure to track eigenvalues near 1 is if they only approach 1 from above. It turns out that this works providing we only take k converging to k0 from below, the proof of which is a key step of the proof of the main result of [EP95]. With all these issues regarding a converse to Theorem 4.1, the true main theorem of [EP95] seems like a reasonable compromise.

4.3 Spectral Duality - Hard Version

Before stating the true main theorem of [EP95], we will use the same conventions of writing the

−2iθj (k) eigenvalues of SD(k) as {e }j∈N, where the eigenphases θj(k) are chosen to lie in [0, π] and satisfy θj ≥ θj+1, which is possible by Theorem 3.7. In keeping with the notation used in [EP95], we start the index j at j = 0. The true main theorem of [EP95] is as follows:

Theorem 4.5 ([EP95], Main Theorem). The following are equivalent:

2 1. −∆D has an M-fold degenerate eigenvalue k0

2. As k ↑ k0, exactly M eigenphases θj(k) of SD(k) converge to π from below.

The second part of this theorem corresponds to the actual eigenvalues converging to 1 in a clockwise direction, in which tracking may be possible due to Theorem 6.9. The proof of this theorem is quite diﬃcult, and relies on a lot of hard analysis. The original takes around 30 pages, so we will only cover the main details. 38 CHAPTER 4. SPECTRAL DUALITY

4.3.1 Potential Theory

The majority of the proof of Theorem 4.5 is based on properties originating in potential theory. For our purposes, potential theory looks at solutions of the Helmholtz equation with given boundary data. We will start by introducing the notation used in [EP95]. The motivation behind these deﬁnitions will be given later, so we can examine the whole picture at once.

We let Γ = ∂D, and s be the arc-length coordinate on Γ, so s ∈ IL = [−L/2, L/2] where L is the 2 2 length of Γ, and we let x : IL → R be the arc length parametrisation of Γ. We let HΓ = L (Γ, ds). 2 c 2 We use ψ, φ for functions on R or D or D = R \D; u, v for functions on Γ. For ease of notation, we’ll write Z Z ψ(x(s))ds = ψ(z)dσ(z). IL Γ We deﬁne the restriction to Γ as:

(γψ)(z) = ψ(z), when z ∈ Γ.

So Z ∗ (γ u, ψ)L2(R2) = u(z)ψ(z)dσ(z). Γ A key property of the restriction is that ker(γ∗) = {0}. Note that for this to make sense we need some regularity assumption on the ψ we are considering. For this, [EP95] uses fractional Sobolev spaces, and lists off a wide variety of regularity properties of the image of γ and its adjoint γ∗. Fractional Sobolev spaces are not something which we have examined, especially not on smooth manifolds, but information may be found in [DNV11]. For this reason, these details will be obfuscated except where absolutely necessary. As a very brief β overview, the Sobolev space HΓ of order β on (Γ, ds) with respect to the arc-length derivative ∂s β 2 1 is defined to be the set of u ∈ HΓ with Λ u ∈ HΓ, where Λ = (1 + (i∂s) ) 2 and ∂s is the arc-length β tangential derivative on Γ. By using Fourier analysis on Γ, HΓ can be thought of as a space of functions β-times weakly differentiable in L2. −1 2 We let G(ζ) = (−∆ − ζ) be the resolvent of the Laplacian on R , and for k > 0 define

G± = lim G(k2 ± i). k ↓0

± Even though we know that −∆ has spectrum (0, ∞), it can be shown the limits defining Gk exist. The + and − in the limits correspond to outgoing and incoming radiation respectively, as can be seen by reintroducing the time dependence as in Section 3.1. The single layer potentials are then defined as Z ± ± Gk u (x) = Gk (x − z)u(z)dσ(z). Γ ± ± ∗ Or equivalently Gk = Gk γ . By definition of G, it then follows that

2 ± ∗ (−∆ − k )Gk u = γ u,

± 2 ∗ ± and so Gk u satisfies the Helmholtz equation on R \Γ. Since ker(γ ) is trivial, so is ker(Gk ). It ± is also shown in [EP95] that Gk u maps HΓ to continuous functions, so we can finally define the 4.3. SPECTRAL DUALITY - HARD VERSION 39

+ + ∗ boundary restriction operator Ak = γGk = γGk γ . This operator forms the backbone of the proof of the main theorem of [EP95]. Due to holomorphicity of G(ζ), Ak is holomorphic in k. That was quite a lot of deﬁnitions without much context, so we’ll now examine the main ideas. Based on our construction of obstacle scattering theory, we want to look at solutions of the 2 Helmholtz equation on R \D. We also want global solutions, and we want them to vanish on Γ. ± 2 To achieve this, we use Gk to map from HΓ to solutions of the Helmholtz equation on R \Γ. If these solutions were then to vanish on Γ, we would be done. So we look back at the boundary + restriction of Gk u, which is exactly the operator Ak. It turns out that from this construction, k is an eigenvalue −∆D if and only if Ak has non-trivial kernel. We state this as a theorem:

± Theorem 4.6 ([EP95], Lemma 5.5). If u ∈ ker(Ak), then ψ = Gk u is an eigenfunction of −∆D with eigenvalue k2. This correspondence is bijective, i.e.

2 dim ker(−∆D − k ) = dim ker(Ak)

+ In fact, since Ak is deﬁned using Gk which corresponds to outgoing radiation, we will be able to use Ak to modify scattering solutions of the free problem to obtain scattering solutions of the problem with obstacle - without modifying the incoming radiation. This is key to the main step in the proof given in [EP95], allowing for the factorisation of the scattering matrix into a form dependant on Ak.

4.3.2 Relating Ak to SD(k)

The ﬁrst thing we’ll do with Ak is decompose it into its real and imaginary parts:

Ak = Yk + iJk, where Yk = <(Ak) and Jk = =(Ak) are self adjoint operators defined in the obvious way. Now we 2 2 2 factorise Jk. For this we need more definitions! We let Fk = {p ∈ R ; |p| = k }, and we’ll use χ to denote functions on Fk. Then we define the restriction to the energy shell Σk via: Z −ip·y (Σkψ)(p) = e ψ(y)dy R2 with adjoint Z ∗ ip·x (Σkχ)(x) = e χ(p)dµ(p), Fk where dµ is the angle measure on the circle Fk, divided by 4π. Note that we could have instead 2 1 defined these operators between functions on R and S by putting the scale factor of k into the ∗ exponents, in which case Σk is exactly the operator H0(k) introduced in Section 3.1. So the same ∗ interpretation holds: Σk maps to solutions of the Helmholtz equation at frequency k. In which case Σkψ can be thought of as extracting the components of ψ at frequency k. We use Σk as opposed to H0(k) to keep with the notation introduced in [EP95].

There are a slew of regularity results concerning Σk, which we will take for granted. Eckmann- ∗ ∗ Pillet then deﬁne Lk = Σkγ . As some motivation, we can see that Lk maps scattering data χ (which 40 CHAPTER 4. SPECTRAL DUALITY

∗ [EP95] take to live on Fk), ﬁrst to the solution Σkχ of the free problem with the given scattering data, and then restrict to Γ. Similarly, Lk maps from boundary data on Γ to corresponding scattering data. This leads to the following Lemmas: 1 √ ∗ 2 p ∗ Lemma 4.7 ([EP95], Lemma 5.1). Jk = πLkLk, and hence Jk = π|Lk| where |Lk| = LkLk is deﬁned via the functional calculus, Theorem 1.18.

∗ Lemma 4.8 ([EP95], Lemma 5.2). The kernels of the operators Ak, A , Jk| 1 and Lk| 1 agree. k − 2 − 2 HΓ HΓ

1 − 2 Remark 4.9. Understanding Lemma 4.8 requires understanding the space HΓ , which is some- ∗ thing we avoid. Essentially, the lemma states that the kernels of Ak and Ak are more regular than 2 merely being L , and agree with the kernels of Jk and Lk, restricted to this more regular space.

This leads us to the key component of the proof in [EP95], which is expressing SD(k) in terms of

Ak. The statement of this theorem has been modiﬁed slightly, as our convention for the scattering matrix at frequency k is that it acts on L2(Sn−1), while [EP95] use the convention that it acts on 2 L (Fk).

Theorem 4.10 ([EP95], Proposition 5.6). For each k > 0, the scattering matrix satisﬁes

−1 ∗ Sk = 1 − 2πiLkAk Lk,

2 2 where Sk : L (Fk) → L (Fk) is given by

(Skχ)(p) = (SD(k)˜χ)(p/k) and χ˜(θ) = χ(kθ).

Note the relationship between Sk and SD(k) is exactly that discussed at the end of Section 3.4.

This distinction between Sk and SD(k) is incurred because [EP95] approach obstacle scattering from the abstract viewpoint.

The point of Theorem 4.10 is that it expresses Sk in a form which depends only on an operator with strong ties to the Dirichlet eigenvalue problem on D, via Theorem 4.6. Also note that a priori −1 2 −1 ∗ Ak may not exist, and certainly doesn’t if k is an eigenvalue of −∆D. The operator Ak Lk can in fact be be defined for all k > 0 in a reasonable way by using regularity conditions on ker(Ak) ∗ −1 and Ran(Lk), and this definition lives up to expectations for the k > 0 at which Ak exists. As Theorem 4.10 is the key step of the method of Eckmann and Pillet, we shall give a rough description of how the result comes about. Suppose we start with some incoming radiation pattern 2 g ∈ L (Fk), with corresponding free scattering solution u0 (up to a factor, such that u0 has the asymptotic form as in Lemma 3.1). We then wish to modify u0 by adding to it some v, such that u0 + v satisfies the Helmholtz equation and vanishes on Γ. So as to not change the incoming radiation pattern, v must only have outgoing part. Hence we want v to be the outgoing solution + operator Gk applied to some f on Γ. For u0 + v to vanish on Γ, it must be that

+ γGk f = Akf = −u0|Γ 4.3. SPECTRAL DUALITY - HARD VERSION 41

−1 Up to a factor of 2πi, due to the scaling in our deﬁnition of u0. Hence f = −2πiAk u0|Γ. This has −1 ∗ outgoing scattering data −2πiLkAk u0|Γ. From the deﬁnition of Lk, it follows that u|Γ = Lkg. So the outgoing scattering data of u0 + v is

−1 ∗ Skg = g − 2πiLkAk Lkg.

Hence −1 ∗ Sk = 1 − 2πiLkAk Lk. This rough heuristic is proven explicitly in [EP95].

We need one more lemma about Ak before we can move onto the proof of Theorem 4.5.

2 Lemma 4.11 ([EP95], Lemma 6.3). For k 6∈ σ(−∆D) there is a unitary operator Uk : HΓ → 2 L (Fk, dµ) for which

Lk = Uk|Lk|.

The proof of this lemma follows from the polar decomposition (see [RS80] for details) combined with the fact that for k as above Lk has trivial kernel, due to Theorem 4.6 and Lemma 4.8.

4.3.3 Proof of Theorem 4.5

We now aim for the proof of Theorem 4.5. To do so, we start by ﬁnding a minimax formula for the eigenphases of Sk.

2 Theorem 4.12 ([EP95], Theorem 6.1). Let k > 0 be such that k 6∈ σ(−∆D). Then for j ∈ N,

(u, Yku) cot θj(k) = inf sup E j+1 u∈EJ+1\{0} (u, Jku) where the inﬁmum is taken over all subspaces Ej+1 ⊂ HΓ of dimension j + 1.

Note that since we start indexing at j = 0 the zeroth eigenvalue will be an inﬁmum over subspaces of dimension 1, which makes sense. To prove Theorem 4.12, we need some more con- 2 structions. For k > 0 with k 6∈ σ(−∆D), we introduce the (inverse) Cayley transform of Sk:

−1 Xk = i(1 + Sk)(1 − Sk) .

2 That this is well-deﬁned follows from Theorem 4.1 and the fact k 6∈ σ(−∆D), so (1−Sk) has dense range and hence Xk has dense domain (noting D(Xk) = Ran(1 − Sk)). There is a subtle point 2 here, in that we require k 6∈ σ(−∆D) for k in some interval to apply the rest of the method of proof. This can be satisﬁed however, since D is bounded with continuous boundary and hence the discussion in Section 2.1 concludes that σ(−∆D) is discrete. It can be checked that unitarity of Sk implies Xk is self adjoint. Then applying the spectral mapping theorem (Theorem 1.19) provides the following lemma:

2 −2iθ Lemma 4.13 ([EP95], Lemma 6.2). Let k > 0 with k 6∈ σ(−∆D). Let θ ∈ (0, π). Then e is an eigenvalue of Sk of degeneracy M if and only if cot θ is an eigenvalue of Xk of degeneracy M. 42 CHAPTER 4. SPECTRAL DUALITY

This lemma combined with Theorem 4.4 then shows that Xk is bounded below and has a ﬁnite number of negative eigenvalues.

With these two facts, we can explain the proof of Theorem 4.12. Since Xk is self adjoint and bounded below, its eigenvalues satisfy the common minimax formula:

(f, Xkf) cot θj(k) = inf sup E0 0 (f, f) j+1 f∈EJ+1\{0} 0 where the infimum is over subspaces Ej+1 ⊂ D(Xk) of dimension j + 1. This minimax formula is −1 ∗ then combined with the representation Sk = 1 − 2πiLkAk Lk of Theorem 4.10 and the factorisa- tions of Lemmas 4.7 and 4.11. There is a lot of simplification and density argument to interchange the set over which the infimum is taken, which then finishes the proof. For notational conve- nience, Eckmann and Pillet only prove this for the smallest eigenvalue j = 0, but their proof easily generalises. We now have enough background material to prove Theorem 4.5.

Sketch of Proof of Theorem 4.5. We ﬁrst prove one implication, that Dirichlet eigenvalues imply 2 convergence of scattering matrix eigenvalues. Fix k0 > 0 such that k0 ∈ σ(−∆D) is an M-fold eigenvalue, and let P be the orthogonal projection onto ker(Ak0 ) ⊂ HΓ. Lemma 4.6 shows that 2 this kernel has dimension M. Now ﬁx u ∈ ker(Ak0 ) and consider k < k0, with k 6∈ σ(−∆D). Due to holomorphicity of Ak, we may write

(u, Aku) = (u, P AkP u) 0 2 = u, P Ak0 + (k − k0)Ak0 + O (k − k0) P u 0 2 2 = (u, P Ak0 P u) + (k − k0) u, P Ak0 P u + O (k − k0) ||u|| 0 2 2 = (k − k0) u, P Ak0 P u + O (k − k0) ||u|| .

It is shown in Lemma 5.4 of [EP95] that P A0 P is a positive operator. Then since ker(A ) is k0 k0 ﬁnite dimensional there exists some C1 > 0 with: 0 2 u, P Ak0 P u ≥ C1||u|| .

So when k < k0, we have 0 2 2 (u, Yku) = <(k − k0) u, P Ak0 P u + O (k − k0) ||u|| 2 2 2 ≤ C1(k − k0)||u|| + O (k − k0) ||u|| and 2 2 (u, Jku) ≥ C2(k − k0) ||u|| for some C2 > 0. So 2 2 2 (u, Yku) C1(k − k0)||u|| + O (k − k0) ||u|| ≤ 2 2 (u, Jku) C2(k − k0) ||u|| C1 = (1 + O (k − k0)) . C2(k − k0) 4.3. SPECTRAL DUALITY - HARD VERSION 43

This converges to −∞ as k ↑ k0, which - combined with Lemma 4.13 and the minimax formula of

Theorem 4.12 - shows that cot θj(k) ↓ −∞ for some j, and hence θj(k) ↑ π. Applying this process for each each u in a basis for ker(Ak0 ) shows that there are at least M eigenphases of Sk converging upwards to π as k ↑ k0.

Now we prove the converse implication. Assume that as k ↑ k0, there are exactly M eigenphases

θj(k) of Sk converging upwards to π. Then by Theorem 4.12 and Lemma 4.13, for each k there exists a subspace Ek ⊂ HΓ of dimension M, such that

(u, Yku) sup ≤ −λk (4.1) u∈Ek\{0} (u, Jku) for some λk which converge to ∞ as k ↑ k0. Now, to avoid introducing another operator that was required for the full proof of the main theorem of [EP95], we will skip some steps and say that Equation 4.1 can be used to produce an orthogonal projection of rank at least M which maps into ker Jk | 1 , and so dim ker Jk | 1 ≥ M. 0 − 2 0 − 2 HΓ HΓ 2 Combined with Theorem 4.6 and Lemma 4.8, this implies that dim ker(−∆D −k ) ≥ M, and hence the converse implication is proven.

Thus Theorem 4.5 is proven.

For the interested reader, the operator we have avoided introducing in the last paragraph of 2 1 the proof is Λ = (1 + (i∂s) ) 2 , where ∂s is the arc-length tangential derivative on Γ, which has many significant impacts on the proofs of all preceding theorems taken from [EP95], and is used in β defining the fractional Sobolev spaces HΓ as mentioned (briefly) in Section 4.3.1.

The proof of Theorem 4.5 was quite technical and very challenging, and gives some interesting insights into any possible generalisations to potential scattering. The main point of the proof was the factorisation of Lemma 4.10, in which the scattering matrix is expressed in a form involving an operator with strong links to the Dirichlet eigenvalue problem on D. While this may not be possible in the potential case, this chapter illuminates many other signiﬁcant spectral properties which may need to be investigated, such as one-sided accumulation and monotone evolution in k. Investigating possible generalisations of Theorem 4.5 will be the main feature of the rest of this thesis. 44 CHAPTER 4. SPECTRAL DUALITY Chapter 5

Generalisations of Spectral Duality to Potential Scattering

In this section, we will develop some possible extensions of the result in [EP95] to potential scattering, for potentials of compact support. This of course has many diﬃculties, as the previous results are relating an “inside” problem and an “outside” problem, with respect to the obstacle under consideration. In the case of potential scattering, it is not clear what the correct generalisation of “inside” and “outside” should be. The result in [EP95] also had an obvious choice of boundary conditions. Before we get to the possible generalisations, we can consider what it actually means for the n Scattering matrix SV (k0) for potential V with compact support in D ⊂ R (an open bounded set with smooth boundary) to have eigenvalue 1. Recalling Remark 4.2, we ﬁnd that the scattering matrix has an eigenvalue 1 at frequency k0 if and only if there exists a scattering solution of 2 (−∆ + V − k0)uV = 0 which agrees identically with a scattering solution of the free equation 2 (−∆ − k0)u0 = 0 outside of D. This leads to a few possible generalisations of the results in [EP95] to potential scattering. Investigating these will be the main aim of this chapter.

5.1 Dirichlet to Neumann Boundary Condition Generalisation

The ﬁrst attempt at a generalisation is as follows. From above, SV (k0) has an eigenvalue 1 if and n 2 only if we have two non-zero functions uV and u0 on all of R satisfying (−∆ + V − k0)uV = 2 (−∆ − k0)u0 = 0, and they agree identically outside of D.

Then u0 restricted to D satisﬁes:

( 2 (−∆ − k0)u0 = 0 on D u0 = uV |∂D on ∂D.

So Λ0(k0)(uV |∂D) = dnu0. But uV and u0 agree up to the boundary of D, so dnu0 = dnu0. So we

ﬁnd that uV satisﬁes the boundary value problem:

45 46 CHAPTER 5. GENERALISATIONS

( (−∆ + V − k2)u = 0 on D 0 V (5.1) dnuV = Λ0(k0)uV |∂D on ∂D Therefore, it might be possible to use this strange boundary value problem to try to generalise the results in [EP95]. Before even considering the nature of solutions as we take k ↑ k0 as in [EP95], we should check that a solution of Equation 5.1 will actually provide us with an eigenvalue 1 for

SV (k0).

Suppose we are given a solution to the above boundary value problemu ˜V , with corresponding n solution to the Helmholtz equationu ˜0. The issue is then ﬁnding a scattering solution on R \D of 2 0 0 (−∆ − k0)˜u = 0 with boundary conditionsu ˜ |∂D =u ˜V |∂D and dnu˜ ≡ −dnu˜V , because this could 2 be used to extendu ˜V to a scattering solution of (−∆ + V − k0)uV = 0 on the whole space (noting the normal vectors for each region point in opposite directions). Similarly, we could extendu ˜0 to 2 a scattering solution of (−∆ − k0)u0 = 0 on the whole space with the same extension, since the boundary value and normal derivatives ofu ˜0 andu ˜V agree.

So we can recover an eigenvalue 1 of SV (k0) if we can ﬁnd a scattering solution of

 (−∆ − k2)˜u0 = 0 on n\D  0 R 0 u˜ =u ˜V on ∂D (5.2)  0  dnu˜ = −dnu˜V . The problem of extending solutions in this way is quite difficult and does not reflect the method given in [EP95], so this first possible extension of the results of [EP95] to potential scattering was not followed any further. Another more computational issue is that both the boundary conditions and the differential operator in the PDE 5.1 depend on k0, which makes comparison for k close to k0 difficult to work with.

5.2 Comparison of Scattering Matrices Generalisation

The next possible generalisation we will cover tries more directly to compare SV (k) and SD(k). In some sense, we wish for scattering from V to look like scattering oﬀ D. Therefore, suppose −1 Sv(k0) SD(k0) has an eigenvalue 1 with eigenfunction f. Then we can say:

SV (k0)f = SD(k0)f = g

2 n−1 for some 0 6= g ∈ L (S ). So by Deﬁnition 3.5, there exist non-zero functions uD, uV deﬁned on n n R \D and R respectively, with the asymptotic form

ik|x| − 1 (n−1) n−1 −ik|x| − 1 (n−1) − 1 (n+1) uD, uV = e |x| 2 f(θ) + i e |x| 2 g(−θ) + O(|x| 2 ), as |x| → ∞ such that 2 (−∆ + V − k0)uV = 0 and 2 (−∆ − k0)uD = 0. 5.3. CAYLEY TRANSFORM GENERALISATION 47

With uD|∂D = 0. But following Remark 4.2, we ﬁnd that uD ≡ uV outside of D. Hence uV |∂D = 0, so uV |D is a Dirichlet eigenfunction of −∆D + V on D with eigenvalue k0 (providing uV |D 6= 0, which we will assume for now for the sake of this heuristic). If there were M linearly independent −1 eigenfunctions of Sv(k) SD(k) with eigenvalue 1, the previous working can be applied to each to obtain M linearly independent Dirichlet eigenfunctions of −∆D + V on D with eigenvalue k0. This leads to the following conjecture:

Conjecture A. The following are equivalent:

−1 1. SV (k) SD(k) has exactly M eigenvalues converging to 1 in a clockwise direction as k ↑ k0.

2. k0 is a Dirichlet eigenvalue of −∆D + V on D, of degeneracy M.

Note that if this proposition were true, then taking V = 0 would return the main result of [EP95] but true in all dimensions, making this an impressive proposition should it be true. −1 This hypothesis had been developed before a solid interpretation of the operator Sv(k) SD(k) had been developed, and after this understanding was achieved it was realised that this was a foolish approach. Physically, the proposition is stating that if the potential asymptotically “looks like” the domain D at frequencies approaching k0, then D must be of a correct shape to support standing waves of the frequency k0. This does not match with the intuition behind the main result of [EP95], which says that an obstacle appears almost asymptotically invisible at frequencies approaching k0 if and only if the obstacle can support standing waves of frequency k0. Another issue is that for −1 SV (k) SD(k) to have an eigenvalue 1, there must be some scattering solution of

2 (−∆ + V − k0)uV = 0

n on R which vanishes identically on ∂D. This seems like quite a diﬃcult condition to impose, given the global nature of the PDE. In fact, by taking V = 0, we retrieve the same issue discussed in Section 4.2, that ∂D must be an analytic set. This is all heuristic of course, and the proposition may be valid. However, this generalisation was abandoned due to these concerns, in preference of the following generalisation.

5.3 Cayley Transform Generalisation

The final and most promising generalisation is as follows. Firstly, recall the Dirichlet to Neumann 2 2 2 operators ΛV (k), Λ0(k) on L (∂D) corresponding to the operators −∆ + V − k and −∆ − k 2 2 respectively. These were defined in Section 2.3, for k such that 0 6∈ σ(−∆D +V −k )∪σ(−∆D −k ). For the moment, we will assume this holds for the values of k we are examining, although we will remove this restriction eventually. Note that since D is bounded with smooth boundary, −∆+V −k2 and −∆ − k2 are both self-adjoint, and L2(∂D) has a complete ONB given by eigenvectors of each Dirichlet to Neumann operator (following Theorems 1.14 and 2.11). So we can introduce the operators −1 −1 CV (k) = (ΛV (k) + i)(ΛV (k) − i) ,C0(k) = (Λ0(k) + i)(Λ0(k) − i) , (5.3) 48 CHAPTER 5. GENERALISATIONS which are well-defined because ΛV (k), Λ0(k) are self adjoint and hence have spectrum lying only on R. One can easily verify that these operators as defined above are both unitary, and so have 2 spectrum lying on the unit circle in C. Due to Theorem 1.18 part (5), L (∂D) has complete ONBs consisting of eigenvectors of CV (k) and C0(k) respectively.

We now form another heuristic argument. Suppose that SV (k0) has eigenvalue 1. For now, we 2 2 will suppose that 0 6∈ σ(−∆D + V − k0) ∪ σ(−∆D − k0), although we will deal with this technicality later on. Then following Remark 4.2, we know that there exist non-zero functions uV , u0 deﬁned n on R such that:

2 2 (−∆ + V − k0)uV = (−∆ − k0)u0 = 0 (5.4) with u0 ≡ uV outside of D. Then we must have u0|∂D = uV |∂D and dnu0 = dnuV , where dn is the outward facing normal derivative on ∂D. Setting g = u0|∂D = uV |∂D (which is non-zero since we 2 2 are assuming 0 6∈ σ(−∆D + V − k0) ∪ σ(−∆D − k0)) and noting u0, uV satisfy Equation 5.4, we ﬁnd that

Λ0(k0)g = dnu0 = dnuV = ΛV (k0)g, and so

(Λ0(k0) + i)g = (ΛV (k0) + i)g.

We can deﬁne f ∈ L2(∂D) by

f := (Λ0(k0) − i)g = (ΛV (k0) − i)g.

But then by combining the previous two equations we ﬁnd

CV (k0)f = C0(k0)f.

−1 So CV (k0)C0(k0) has eigenvalue 1 with f as eigenfunction (noting f cannot be zero, as (Λ0(k0)−i) is invertible by self-adjointness of Λ0(k0), and g 6= 0 as mentioned previously). −1 So, we know that if SV (k0) has eigenvalue 1 then CV (k0)C0(k0) must have eigenvalue 1 also, up 2 2 to the issue of maintaining 0 6∈ σ(−∆D + V − k0) ∪ σ(−∆D − k0). By running this same procedure for M linearly independent eigenvectors of SV (k0) with eigenvalue 1, we can produce M linearly −1 independent eigenvectors of CV (k0)C0(k0). This leads to our most promising generalisation of the results of [EP95]:

Conjecture B. The following are equivalent:

−1 1. CV (k)C0(k) has exactly M eigenvalues converging to 1 in a clockwise direction as k ↑ k0.

2. SV (k) has exactly M eigenvalues converging to 1 in a clockwise direction as k ↑ k0

We also have the simpler form, in the case the eigenvalue 1 is actually attained.

−1 Conjecture C. If SV (k0) has 1 as an eigenvalue of degeneracy M, then CV (k)C0(k) has eigenvalue 1 of degeneracy M. 5.3. CAYLEY TRANSFORM GENERALISATION 49

It will become apparent as we try to prove this that many assumptions need to be made about V , and even the type of potential scattering we consider will need to change to one with more influence from the potential as k → ∞. These adjustments will be made as we find them −1 necessary. We will also see later that CV (k0)C0(k0) actually attains eigenvalue 1 whenever the above holds, although passing between approaching frequencies k and the desired frequency k0 may be helpful (this result depends on some assumptions about limiting behaviour of the eigenvalues of −1 CV (k)C0(k) as k ↑ k0). Before going into more research about the validity of ConjectureB, some computational ex- −1 amples were checked. The eigenvalues of CV (k)C0(k) were computed explicitly for D the ball of radius 1 centred at the origin, and V the characteristic function of the ball of radius half multiplied by a constant (which was varied to examine dependence). Upon completing these computational −1 examples, it was found that the spectrum of CV (k)C0(k) did not move in a way that seemed −1 conducive of the intended results. Namely, eigenvalues of CV (k)C0(k) would sit at or near 1, then move around the unit circle slightly when k was varied, always returning to near 1 and never completing full laps of the unit circle (see Figure 5.1). After some discussion, it was realised that this may be an issue with how the original problem was posed, and how it compares to the result for obstacle scattering. In obstacle scattering, the obstacle and boundary conditions play an important role at all frequencies k, while for potential scattering the potential hardly affects solutions as k → ∞. For this reason, we slightly modify our problem.

−1 Figure 5.1: The dark arc is an arc of the unit circle in C. Eigenvalues of CV (k)C0(k) would follow paths similar to that drawn in dashed grey.

We will from here on consider semi-classical scattering, in which we consider −∆+k2V −k2 as the perturbed operator. This is no longer directly related to eigenvalue problems, but in this modified regime the potential “k2V ” is significant at any frequency k. We will also use the semi- classical Dirichlet to Neumann operators, which differ from the classical Dirichlet to Neumann 50 CHAPTER 5. GENERALISATIONS operators of Chapter2 only in that they are divided by the frequency. I.e, they have graph 1 {(f, g) ∈ L2(∂D) × L2(∂D); ∃u ∈ H1(D), −∆u + k2V u − k2u = 0, u|∂D = f, d u = g}. k n It can be verified that all the theorems applicable to the classical scattering matrix and Dirichlet to Neumann operators are satisfied by their semi-classical counterparts, with extra factors of k scattered throughout the proofs. For this reason, from here on all our operators will be semi- classical rather than classical.

We will use the bold-face Λ0,V (k) to denote the semiclassical Dirichlet to Neumann operators, and C0,V (k) to denote Cayley transform of the semi-classical Dirichlet to Neumann operators, as in Equation 5.3.

5.3.1 Occurrence in Literature, and Notational Conventions

−1 After some time investigating properties of the operator CV (k)C0(k) it was discovered that this operator has some connections to previous literature. Namely, a paper of Kirsch and Lechleiter, [KL13]. This paper studies a possible potential scattering generalisation of the results of [EP95]. Before detailing their main result, we have the following definition (slightly modified from the definition used in [KL13]) :

Deﬁnition 5.1 ([KL13], Deﬁnition 2.2). k0 > 0 is an interior transmission eigenvalue if there 2 2 exists a non-trivial pair uV , u0 ∈ L (D) × L (D) such that

2 2 2 (−∆ + k0V − k0)uV = (−∆ − k0)u0 = 0 in D and uV |∂D = uV |∂D, dnuV = dnu0.

The properties of interior transmission eigenvalues is a topic which has been studied in some depth by mathematical physicists working in inverse problems, and they have their own set of notation different to that used in [EP95]. Also note that the interior transmission eigenvalue problem is identical to the proposed generalisation of Section 5.1 by definition of the Dirichlet to Neumann operator, after the change to the semiclassical equation. The main result of [KL13] was (up to some simplification):

Theorem 5.2 ([KL13], Theorem 6.3). Let k0 > 0 and I = (k0 − , k0 + )\{k0} such that no k ∈ I −2iδn(k) is a transmission eigenvalue. For k ∈ I, let µn(k) = e be the eigenvalues of SV (k) with phases δn(k) ∈ (0, π).

1. Let k0 be the smallest interior transmission eigenvalue with V < 0. Then

lim δ−(k) = 0, (5.5) k∈I,k↑k0

where δ−(k) = minn δn(k).

2. Let 5.5 hold and V < 0 in D. Then k0 is an interior transmission eigenvalue. 5.3. CAYLEY TRANSFORM GENERALISATION 51

3. Let k0 be the smallest interior transmission eigenvalue with V > 0. Then

lim δ+(k) = π, (5.6) k∈I,k↑k0

where δ+(k) = maxn δn(k).

4. Let 5.6 hold and V > 0 in D. Then k0 is an interior transmission eigenvalue.

The issues with V > 0 and V < 0 are due to the accumulation of eigenvalues of SV (k) at 1 from either above or below. This would be quite a good result even if it only holds for the smallest interior transmission eigenvalue. However, the proof of this theorem was only completed with very stringent conditions on V : V must be constant, and either positive and sufficiently large, or negative but sufficiently close to 0. This is quite a blow to the usefulness and generality of this theorem. The majority of the proof of this theorem followed similar methods to that used in [EP95] and explained in Chapter4, namely the scattering matrix (in fact SV (k) − I) was factorised, with one factor an operator characterising interior transmission eigenvalues. A minimax formula was then used for the maximum and minimum eigenphases of the scattering matrix, given in terms of interior information. See [KL13] for details of the proof. Despite the result of [KL13] only having been proven in very restrictive cases, it does provide some hope for our latest generalisation. It can be seen by direct comparison that k0 > 0 is an −1 interior transmission eigenvalue if and only if CV (k0)C0(k0) has eigenvalue 1. For this reason, we can finally give our operator a fitting name:

Deﬁnition 5.3. The transmission operator at frequency k > 0 on domain D is the operator −1 CV (k)C0(k), where CV (k), C0(k) are as above. Dependence on the domain may be added to −1 notation as in CD,V (k)CD,0(k), and k may be dropped from the notation.

At this point of this thesis, the above deﬁnition only makes sense for k such that 0 6∈ σ(−∆D + 2 2 2 k V − k ) ∪ σ(−∆D − k ). This condition will be removed shortly, and the resulting operator will still be referred to as the transmission operator. In fact, it will be seen that this family of operators depends holomorphically on k in some neighbourhood of the real axis, and that it has many spectral properties similar to those of the scattering matrix (under some reasonable assumptions on V ). This seems to suggest that studying the transmission operator, as opposed to the interior transmission eigenvalue problem, will generate stronger links to the scattering matrix. 52 CHAPTER 5. GENERALISATIONS Chapter 6

−1 Properties of CV (k)C0(k)

n Throughout this chapter, D will be an open bounded subset of R with smooth boundary, and unless otherwise stated transmission operators will be on domain D. BR will be a ball centred at the origin, of radius large enough to contain the support of V (see Figure 6.1).

B R D

supp(V)

Figure 6.1: The inner region is supp(V ), the next curve out from the centre is the boundary of D, and the next line out again is the boundary of BR. This is for the case n = 2, which we do not assume in general.

−1 This chapter will start by covering some basic properties of the transmission operator CV (k)C0(k). These basic properties will be used to prove ConjectureC in the central potential case. Next, a result is proven which shows that we may just consider transmission operators on domains which are balls without losing too much generality. Three important spectral properties of the transmission operator are then verified. These concern the accumulation of eigenvalues, the flow of eigenvalues as frequency is varied, and the attainment of eigenvalue 1. 2 The first objective of this section is removing the issues about ensuring 0 6∈ σ(−∆D + k V −

53 −1 54 CHAPTER 6. PROPERTIES OF CV (K)C0(K)

2 2 −1 k ) ∪ σ(−∆D − k ). It turns out that the transmission operator CV (k)C0(k) can be defined for all k in a neighbourhood of the real axis in C, and that this operator varies holomorphically in k. Moreover, the eigenvalues and eigenspaces vary holomorphically away from accumulations, so their dynamics can be explored using “standard” calculus. This is achieved through Theorems proven in [BH14], given some slight modification. The proof in [BH14] actually requires the domain to be strictly star-shaped with respect to the origin, although this was only necessary for a certain coefficient to be positive. This coefficient has been removed in our modified theorem, and so the star-shaped condition can be dropped (the proof still works as in [BH14]). The composite result is as follows:

∞ Theorem 6.1. Assume V ∈ Cc (D) has non-negative real part and is bounded away from the value 1. Then there exists a neighbourhood of the real axis U ⊂ C such that for all k in U and f ∈ L2(∂D), there is a unique solution to the problem

1 (−∆ + k2V − k2)u = 0 in D, d u − iu| = f. k n ∂D Furthermore, u(k) depends holomorphically on k ∈ U. The mapping f 7→ u is linear and bounded as a map L2(∂D) → H1(D).

Remark 6.2. We will almost always require V to be real valued, in which case the previous theorem shows that even though the operator ΛV (k) might not make sense we can always define an operator −1 (ΛV (k) − i) . For k such that ΛV (k) is defined, this will be the same as the inverse of ΛV (k) − i, and the extension works so that this family depends holomorphically on k in some neighbourhood −1 of the real axis. Theorem 2.13 shows that (ΛV (k) − i) is compact wherever ΛV (k) is defined. 2 In fact, since D is bounded with continuous boundary the spectrum of −∆D + k V is discrete

(following the discussion in Section 2.1), so ΛV (k) is deﬁned everywhere except for a discrete set.

By the holomorphicity of the extension to all k ∈ U, for any k0 ∈ U with ΛV (k0) not defined we can −1 −1 take a sequence kn → k0 in U with ΛV (kn) defined, and find that (ΛV (kn)−i) → (ΛV (k0)−i) −1 in norm. So following the discussion in Section 1.1 regarding compact operators, (ΛV (k0) − i) is −1 1 compact. Alternatively, (ΛV (k0) − i) f is the trace of the u ∈ H (D) as in Theorem 6.1. Trace −1 is compact (by Theorem 2.9) and f 7→ u is bounded, so (ΛV (k0) − i) is compact. In conclusion, −1 there exists some neighbourhood U of R ⊂ C such that (ΛV (k) − i) depends holomorphically on k, and is a compact operator for each k. The proof in [BH14] can be repeated with the minus swapped to a plus, in which case we obtain −1 the same results for (ΛV (k) + i)

Corollary 6.3. The semi-classical Cayley transform CV (k) can be deﬁned for all k ∈ U ⊂ C for some neighbourhood U of the real axis. It is deﬁned by:

1 1 f = d u − iu| 7→ d u + iu| , k n ∂D k n ∂D where u ∈ H1(D) satisﬁes (−∆ + k2V − k2)u = 0 in D. By elliptic regularity (see [Eva10]), u is in fact smooth in the interior of D. 55

2 L (∂D) has a complete orthonormal basis of eigenvectors of CV (k) for each k, and CV (k) is unitary for real k.

Furthermore, in any interval I of the unit circle in which the spectrum of CV (k) is discrete at some k = k0, the eigenvalues and eigenspaces for eigenvalues in I depend holomorphically on k in some neighbourhood of k0. The requirement that V has non-negative real part is the second adaptation to the Main The- oremB after the move to semi-classical scattering. The condition is used so that the inverse of

−∆D + V is defined and compact, which is necessary for the proof of Theorem 6.1 as given in [BH14]. That V is bounded away from the value 1 is required in the modified proof of Lemma A.3 of [BH14]. Positivity of the real part of V seems like a natural assumption to take in regards to the main theorem of [EP95], since an obstacle could be thought of as a compactly supported potential which takes positive and infinitely large values on its support. Issues of avoiding taking the value 1 come from the semiclassical problem, in which we consider the operator −∆ + k2V − k2, which collapses to the Laplacian wherever V = 1.

The corollary follows directly from the theorem and the fact that ΛV (k) is bounded below, self adjoint and has compact resolvent whenever it exists (Theorem 2.13), so gives a complete 2 decomposition of L (D) into eigenspaces of ΛV (k), by Theorem 1.14. Then by the spectral theorem,

Theorem 1.19, CV (k) also gives a complete eigenspace decomposition whenever ΛV (k) is deﬁned. Using the holomorphic dependence on k, we can extend this eigenspace decomposition to the 2 diﬃcult values of k. This allows us to stop being so careful with the condition 0 6∈ σ(−∆D + k V − 2 2 k ) ∪ σ(−∆D − k ). In fact, by the uniqueness result of Theorem 6.1, we see that 1 1 f := d u − iu | = d u − iu | k n 0 0 ∂D k n V V ∂D in our discussion motivating TheoremB cannot possibly be zero, so the requirement 0 6∈ σ(−∆D + 2 2 2 k V − k ) ∪ σ(−∆D − k ) is not necessary (the extra factors of k due to the introduction of the semi-classical problems don’t actually change anything regarding uniqueness, as can be seen in the proof of this theorem in [BH14]). To be able to even consider proving some form of ConjectureB, we need to know about how −1 the eigenvalues of CV (k)C0(k) change with variation in k, and how they are accumulated. Before −1 getting into this, we’ll characterise the eigenvalues of CV (k)C0(k) ∞ −1 Theorem 6.4. Assume V ∈ Cc (D) is non-negative and bounded away from the value 1. CV (k)C0(k) iθ 1 has eigenvalue e if and only if there exist uV , u0 non-zero in H (D) and smooth in the interior, such that 2 2 2 (−∆ + k V − k )uV = (−∆ − k )u0 = 0 in D and 1 1 1 1 d u − iu | = d u − iu | , d u + iu | = eiθ d u + iu | . k n 0 0 ∂D k n V V ∂D k n 0 0 ∂D k n V V ∂D

2 −1 iθ If f ∈ L (∂D) is the eigenvector of CV (k)C0(k) with eigenvalue e , then 1 1 f = d u − iu | = d u − iu | . k n 0 0 ∂D k n V V ∂D −1 56 CHAPTER 6. PROPERTIES OF CV (K)C0(K)

iθ Furthermore, the eigenvalue e has degeneracy M if and only if M such pairs (uV , u0) exist and are linearly independent.

2 −1 iθ Proof. Let f ∈ L (∂D) be an eigenvector of CV (k)C0(k) with eigenvalue e . Then by Corollary 1 6.3 there exists uV , u0 non-zero in H (D) and smooth in the interior such that

2 2 2 (−∆ + k V − k )uV = (−∆ − k )u0 = 0 and 1 1 f = d u − iu | = d u − iu | . k n V V ∂D k n 0 0 ∂D The eigenvalue condition can be read as

iθ C0(k)f = e CV (k)f.

Using the deﬁnition of the Cayley transforms, we obtain 1 1 d u + iu | = eiθ d u + iu | . k n 0 0 ∂D k n V V ∂D 1 1 Alternatively, if given u0, uV as above, then f = k dnu0 − iu0|∂D = k dnu0 − iu0|∂D will be such −1 iθ that CV (k)C0(k)f = e f. To see that f is non-zero we apply Theorem 6.1. To prove the statement about degeneracy of eigenspaces, just repeat the above with each vector in a basis of the eigenspace.

There is a simple and helpful corollary:

−1 1 Corollary 6.5.C V (k)C0(k) has eigenvalue 1 if and only if there exist uV , u0 in H (D) and smooth in the interior such that

2 2 2 (−∆ + k V − k )uV = (−∆ − k )u0 = 0, with dnu0 = dnuV and u0|∂D = uV |∂D. This is proven by solving the simultaneous equations presented in Theorem 6.4 for eiθ = 1.

6.1 The Central Potential Case

In the case that V (x) ≥ 0 is a central potential (depends only on r = |x|) we can actually show that ConjectureC holds directly for domain some open ball BR centred at the origin containing supp(V ), and in fact holds as an if and only if statement. We have already seen in in Section 5.3 −1 that if SV (k0) has 1 as an M-fold degenerate eigenvalue, then CV (k)C0(k0) has 1 as an eigenvalue −1 of degeneracy at least M. So, suppose instead that CV (k0)C0(k0) has 1 as an eigenvalue of 2 degeneracy M, with f ∈ L (∂BR) as an eigenvector. Then by the corollary to Theorem 6.4, there 1 exists u0, uV in H (BR) and smooth in the interior such that  (−∆ − k2)u = (−∆ + k2V − k2)u = 0 on B  0 0 0 0 V R u0 = uV on ∂BR   dnu0 = dnuV . 6.1. THE CENTRAL POTENTIAL CASE 57

We will see that u0, uV can be extended to be equal outside of BR while still satisfying the n above PDE (with BR replaced by R ), thus obtaining an eigenfunction of SV (k0) with eigenvalue 1 by the Remark 4.2. Since both u0 and uV are smooth, they can be decomposed into a sum of spherical harmonics multiplied by smooth functions of r, i.e.

∞ ∞ X X u0(rθ) = R0,i(r)Θi(θ), uV (rθ) = RV,i(r)Θi(θ), i=0 i=0

x n−1 where r = |x|, θ = |x| and Θi(θ) are the spherical harmonics on S . See Section 2.1.1 for details. 2 n−1 Since {Θi(θ)} form an ONB for L (S ), it suﬃces to show that each pair R0,i(r)Θi(θ),RV,i(r)Θi(θ) n can be extended to all of R in such a way that they agree outside of BR. Writing out our required PDE conditions in spherical coordinates and using Equation 2.3 gives:

2 2 0 = −∆ + k0V − k0 RV,i(r)Θi(θ) (n − 1) 1 = −∂2 − ∂ − ∆ + k2V − k2 R (r)Θ (θ) r r r r2 tan 0 0 V,i i (n − 1) 1 = Θ (θ) −∂2R (r) − ∂ R (r) + k2VR (r) − k2R (r) − R (r) ∆ Θ (θ) i r V,i r r V,i 0 V,i 0 V,i V,i r2 tan i (n − 1) 1 = Θ (θ) −∂2R (r) − ∂ R (r) + k2VR (r) − k2R (r) − R (r) C Θ (θ) i r V,i r r V,i 0 V,i 0 V,i V,i r2 i i 1 = Θ (θ) −r2∂2R (r) − (n − 1)r∂ R (r) + k2V r2R (r) − k2r2R (r) − C R (r) , r2 i r V,i r V,i 0 V,i 0 V,i i V,i

where Ci is the eigenvalue of ∆tan for the spherical harmonic Θi(θ). This is satisﬁed if we can solve

2 2 2 2 2 2 0 = −r ∂r RV,i(r) − (n − 1)r∂rRV,i(r) + k0V r RV,i(r) − k0r RV,i(r) − CiRV,i(r).

Similarly, by removing V from the above calculations we ﬁnd

2 2 2 2 0 = −r ∂r R0,i(r) − (n − 1)r∂rR0,i(r) − k0r R0,i(r) − CiR0,i(r).

To satisfy u0 = uV on ∂BR and dnu0 = dnuV , we need

R0,i(R) = RV,i(R), (∂rR0,i(r))|r=R = (∂rRV,i(r))|r=R.

So, our problem of extension comes down to solving the ODEs

( 2 2 2 2 2 2 −r ∂r X(r) − (n − 1)r∂rX(r) + k0V r X(r) − k0r X(r) − CiX(r) = 0 on [R, ∞) 0 X(R) = RV,i(R),X (R) = (∂rRV,i(r))|r=R, and ( 2 2 2 2 −r ∂r X(r) − (n − 1)r∂rX(r) − k0r X(r) − CiX(r) = 0 on [R, ∞) 0 X(R) = R0,i(R),X (R) = (∂rR0,i(r))|r=R, and showing the solutions agree asymptotically. But since V is supported on BR and the initial conditions match, these ODEs are actually identical. It is shown in [Eva10] that such ODE always n have solutions which are unique. Hence we can extend R0,i(r)Θi(θ),RV,i(r)Θi(θ) to all of R such −1 58 CHAPTER 6. PROPERTIES OF CV (K)C0(K)

2 2 2 that the extensions are annihilated by (−∆ − k0) and (−∆ + k0V − k0) respectively, and they 2 agree for r ≥ R. Summing over i, we obtain extensions of u0, uV annihilated by (−∆ − k0) and 2 2 (−∆ + k0V − k0) respectively, such that u0 = uV outside of BR. Due to Remark 4.2, SV (k0) must have 1 as an eigenvalue. Repeating this process for M linearly independent eigenvectors of −1 CV (k0)C0(k0) with eigenvalue 1 produce M linearly independent eigenfunctions of SV (k0) with eigenvalue 1. Combined with the earlier result that SV (k0) having 1 as an M-fold degenerate −1 eigenvalue implies that CV (k)C0(k0) has 1 as an eigenvalue with eigenspace of dimension at least M, we see that TheoremC is proven completely in the case that V is radial.

6.2 Invariance Under Change of Domain

In this subsection, we’ll prove a nice theorem which shows that we may restrict to considering −1 CV (k)C0(k) on a ball BR centred at the origin without losing too much information, while working ∞ n on the ball will be helpful for calculations later on. Fix some non-negative V ∈ Cc (R ) with values bounded away from 1 and let supp(V ) ⊂ D ⊂ BR where D is strictly contained in BR.

∞ Theorem 6.6. Assume V ∈ Cc (D) is non-negative and bounded away from the value 1. If C−1 (k)C (k) has eigenvalue 1 with degeneracy M, then C−1 (k)C (k) has eigenvalue 1 BR,V BR,0 D,V D,0 with degeneracy at least M.

1 Proof. By Theorem 6.4, there exists M non-zero linearly independent pairs uV , u0 in H (BR) which are smooth in the interior, such that

2 2 2 (−∆ + k V − k )uV = (−∆ − k )u0 = 0 in BR and

uV |∂BR = u0|∂BR , dnuV = dnu0.

If we let w = uV − u0, then noting that supp(V ) ⊂ D by assumption, w solves

2 (−∆ − k )w = 0 on BR\D and

w|∂BR = dnw|∂BR = 0.

We claim this implies w ≡ 0 on BR\D. To see this, ﬁrst note that since D is strictly contained in BR and both are open, there is some smaller ball BR0 such that D ⊂ BR0 ⊂ BR. Then w solves the same PDE on BR\BR0 , and so we can write w in terms of spherical harmonics on the spherically symmetric domain BR\BR0 ,

∞ X 0 w = Ri(r)Θi(θ), for R < r < R, i=1 where each Ri is smooth since uV and u0 are (see Theorem 2.4 for details). By linear independence 0 of the {Θi} and the boundary conditions on w, each Ri must satisfy Ri(R) = Ri(R) = 0. But by separating variables and recalling the comments after Theorem 2.4, each Ri satisﬁes some second 6.3. SPECTRAL PROPERTIES 59 order linear diﬀerential equation to which existence and uniqueness apply (see [Eva10]), so by uniqueness Ri ≡ 0. Hence w ≡ 0 on BR\BR0 . By Theorem 3.2 of [CC06], solutions to the Helmholtz equation (−∆ − k2)v = 0 are real analytic. But w vanishes on the open and non-empty set BR\BR0 , so w must vanish on all of

BR\D. So uV = u0 on BR\D, and so the boundary value and normal derivatives of uV |D, u0|D on D agree. Since u0 is nonzero in BR and solves the Helmholtz equation, u0 is real analytic and hence cannot vanish on any non-empty open set in BR. So u0|D, uV |D are a nontrivial pair −1 satisfying the requirements of Theorem 6.4, and hence CD,V (k)CD,0(k) has eigenvalue 1. This can be repeated for each of the M linearly independent pairs uV , u0 to obtain M linearly independent −1 −1 eigenfunctions of CD,V (k)CD,0(k) of eigenvalue 1. Hence CD,V (k)CD,0(k) has 1 as an eigenvalue of degeneracy at least M, as claimed.

It is important to note that this only works in one direction: we can pass to smaller domains while maintaining existence of eigenvalue 1, but we can not extend to larger domains and hope to maintain eigenvalue 1. This would require being able to extend uV and u0 to the larger domain, which is in general not possible. However, we’ve seen that such extension is possible in the case that V is a central potential, but this assumption is very strong.

6.3 Spectral Properties

−1 We will start by covering some of the simplest spectral properties of CV (k)C0(k), before getting into the tougher parts. We have the following ﬁrst step:

∞ Theorem 6.7. Assume V ∈ Cc (D) is non-negative and bounded away from the value 1. For each −1 k in some neighbourhood of R ⊂ C, CV (k)C0(k) = I + K(k) where K is compact for each k and depends holomorphically on k.

Proof. We start with k such that ΛV (k) and Λ0(k) are deﬁned. Then

−1 C0(k) = (Λ0(k) + i)(Λ0(k) − i) −1 = I + 2i(Λ0(k) − i) .

By the remark following Theorem 6.1, this in fact must hold for all k in a neighbourhood of R ⊂ C, −1 and (Λ0(k) − i) is compact and depends holomorphically on k in this neighbourhood. Similarly, −1 −1 −1 −1 CV (k) = (ΛV (k) − i)(ΛV (k) + i) = I − 2i(ΛV (k) + i) , with (ΛV (k) + i) compact and holomorphic on some neighbourhood of R ⊂ C. Then

−1 −1 −1 CV (k)C0(k) = I − 2i(ΛV (k) + i) I + 2i(Λ0(k) − i) −1 −1 −1 −1 = I − 2i(ΛV (k) + i) + 2i(Λ0(k) − i) + 4(ΛV (k) + i) (Λ0(k) − i) ,

−1 −1 which is of the form I + K(k) for K(k) = −2i(ΛV (k) + i) + 2i(Λ0(k) − i) + 4(ΛV (k) + −1 −1 i) (Λ0(k) − i) which is compact and holomorphic on some neighbourhood of R ⊂ C as each factor and summand is compact and holomorphic on some neighbourhood of R ⊂ C. −1 60 CHAPTER 6. PROPERTIES OF CV (K)C0(K)

We can then use some of the wonderful properties of compact operators (as discussed in Section −1 1.1, such as Theorem 1.15), along with holomorphicity and unitarity of CV (k)C0(k) to obtain the following Lemma:

∞ Corollary 6.8. Assume V ∈ Cc (D) is non-negative and bounded away from the value 1. Then −1 for each k in a neighbourhood of R ⊂ C, the transmission operator CV (k)C0(k) has spectrum consisting only of eigenvalues. These eigenvalues have unit norm, ﬁnite degeneracy, and accumulate at 1 only. Eigenvalues and eigenvectors may be chosen to depend holomorphically on k away from the accumulation at 1.

Now we can start to work on the more diﬃcult spectral properties that we need to investigate.

6.3.1 Accumulation of Eigenvalues

One important property of the scattering matrix which was required for the proof of the main theorem of [EP95] is that the spectrum accumulates at 1 from only one side, and eigenvalues move away from the accumulation back around the unit circle towards 1. It has been shown (see for example [GRH12]) that eigenvalues cannot typically be tracked continuously through an accumulation. Since the result we are aiming for crucially depends on tracking eigenvalues near 1, −1 we need to know that the spectrum of CV (k)C0(k) accumulates at 1 from only one side. −1 To do this, we will consider for t ∈ [0, 1] the operator CtV (k)C0(k) for fixed k (we will actually need to consider t in a neighbourhood of [0, 1] ⊂ C). As we increase t from zero to one, we are essentially “turning on” the potential. We know that at t = 0 this operator is just the identity, with −1 spectrum all at 1, and at t = 1 it is CV (k)C0(k), whose spectrum we wish to investigate. We’ll aim to show that as t is increased, eigenvalues move around the unit circle in one direction. Before −1 getting to that, we need to know that the spectrum of CtV (k)C0(k) only consists of eigenvalues. This can be seen by slightly modifying the proof of Theorem 6.7 and Corollary 6.8. The −1 eigenvalues and eigenvectors of CtV (k)C0(k) can be chosen to depend holomorphically on t in a neighbourhood D ⊂ C of [0, 1], by resorting to a modified form of Theorem 6.3 and its following discussion. Note that this requires the result of Theorem 6.3 to hold for complex potentials, as tV may have non-zero imaginary part. Thus it is not a coincidence then that Theorem 6.3 was explained in the context of smooth compactly supported potentials with requirements only on their real part, although this is the only time in this whole thesis that we will actually consider non-real −1 potentials. By a slight modification of Theorem 6.7, we can see that CtV (k)C0(k) is a compact −1 perturbation of the identity, CtV (k)C0(k) = I + K(t) for K compact and holomorphic in t on D.

∞ Theorem 6.9. Assume V ∈ Cc (D) is non-negative and bounded away from the value 1. Then −1 the eigenvalues of CV (k)C0(k) accumulate at 1 only, and from below. I.e there are only ﬁnitely many eigenvalues with positive imaginary part.

We have already seen that the eigenvalues can only accumulate at 1, by the corollary to Theorem 6.7. The proof of the second claim of this theorem is quite a task, and requires a few lemmas. First −1 we introduce some notation common to all parts of the proof. We set U(t) := CtV (k)C0(k) (which is unitary for real t, and is a compact perturbation of the identity), and we will use {eiβn(t)} for the 6.3. SPECTRAL PROPERTIES 61 set of eigenvalues of U(t) with corresponding ONB of eigenvectors {gn(t)}. These can be chosen to depend holomorphically on t in a neighbourhood Ω ⊂ C of [0, 1], with βn(0) = 0 and ||gn(t)|| = 1, iβn(t) until e → 1 in which case we continuously extend βn(t) to be constant. Dots above functions will represent derivatives with respect to t.

Lemma 6.10. β˙n(t) ≤ 0 for t ∈ [0, 1].

iβn(t) Proof. Suppose e is an eigenvalue of U(t) with eigenfunction gn(t) with ||gn(t)||L2(Sn−1) = 1. iβn(t) We only consider t for which e is away from 1, since otherwise βn(t) is constant and the result holds trivially. Since t is real, βn(t) is real by unitarity of U(t). From here we drop t and n from the notation. Then eiβg = U(t)g.

Diﬀerentiating, taking inner product with Ug = eiβg and noting unitarity of U(t) and realness of β for t ∈ [0, 1] gives

iβe˙ iβg + eiβg˙ = Ug˙ + Ug˙ iβ˙ + eiβg,˙ eiβg = Ug,˙ eiβg + (Ug,˙ Ug) iβ˙ + (g, ˙ g) = Ug,˙ eiβg + (g, ˙ g) iβ˙ = Ug,˙ eiβg .

(6.1)

−1 For U(t) = CtV (k)C0(k), we have

˙ −1 ˙ −1 −1 ˙ U = CtV CtV CtV (k)C0(k) = CtV CtV U.

So: iβ˙ = Ug,˙ eiβg

−1 ˙ iβ = CtV CtV Ug, e g iβ −1 ˙ iβ = e CtV CtV g, e g −1 ˙ = CtV CtV g, g = C˙tV g, CtV g . (6.2)

By Theorem 6.3, let us now introduce wt the solution of:

2 2 (−∆ + k tV − k )wt = 0 on D 1 k dnwt − iwt|∂D = g on ∂D, with 1 C g = d w + iw | . tV k n t t ∂D −1 62 CHAPTER 6. PROPERTIES OF CV (K)C0(K)

Note that by holding g constant we find: 1 C˙ g = d w˙ + iw˙ | , (6.3) tV k n t t ∂D wherew ˙t satisfies: d 2 2 2 2 2 0 = dt (−∆ + k tV − k )wt = (−∆ + k tV − k )w ˙t + k V wt on D 1 k dnw˙t − iw˙t|∂D = 0 on ∂D. See Section 1.2 for details on differentiating operator valued functions.

Note w˙t satisﬁes the same equation inside D with wt replaced by wt, since we assume V and k are real. Using the boundary conditions onw ˙t and Equation 6.3, we actually see: 1 C˙ g = 2 d w˙ = 2iw˙ | . tV k n t t ∂D Inserting into Equation 6.2 gives: ˙ iβ = C˙tV g, CtV g 1 = C˙ g, d w + C˙ g, iw | tV k n t tV t ∂D 1 1 = 2iw˙ | , d w + 2 d w˙ , iw | t ∂D k n t k n t t ∂D 2i = ((w ˙ | , d w ) − (d w˙ , w | )) k t ∂D n t n t t ∂D 2i Z = (w ˙tdnwt − wtdnw˙t) . k ∂D

Using Green’s Formula (Theorem 2.1) gives: 2 Z β˙ = (w ˙tdnwt − wtdnw˙t) k ∂D 2 Z = (w ˙t∆wt − wt∆w ˙t) . k D

Finally, adding and subtracting terms, and using the PDE which are satisﬁed by wt andw ˙t gives: 2 Z β˙ = (w ˙t∆wt − wt∆w ˙t) k D Z 2 2 2 2 2 = − w˙t(−∆ + k tV − k )wt − wt(−∆ + k tV − k )w ˙t k D Z 2 2 = − wtk V wt (6.4) k D Z 2 = −2k V |wt| D ≤ 0, (6.5) as claimed. 6.3. SPECTRAL PROPERTIES 63

Lemma 6.11. {βn(t)} is equicontinuous.

Proof. Note from Equation 6.4, we have: Z ˙ 2 2 |βn(t)| ≤ 2k |V ||wt,n| ≤ 2kC||wt,n||L2(D), D 2 where C > 0 is some bound on |V |. Since the {gn} are uniformly bounded in L (∂D) (each has norm

1) and the mapping gn → wn,t is bounded by Theorem 6.1, we ﬁnd that |β˙n(t)| is bounded independently of n, wherever β˙n exists. Wherever β˙n does not exist, βn has been continuously extended to be constant. It is trivial to show that derivative bounds such as these imply equicontinuity. So we ﬁnd that {βn(t)} is equicontinuous as claimed. Note we only prove pointwise equicontinuity, so nice dependence on t is not assumed.

Lemma 6.12. Suppose {fn} is a countable equicontinuous family of functions [0, 1] → R, and {tn} ⊂ [0, 1] is a sequence converging to t0 ∈ [0, 1]. Suppose fn(tn) ≤ 0 for all n. Then for all > 0, inﬁnitely many n ∈ N satisfy fn(t0) < .

Proof. Fix > 0. By equicontinuity of {fn} at the point t0, there exists a δ > 0 such that for all n ∈ N and all t ∈ [0, 1] with |t − t0| < δ we have |fn(t0) − fn(t)| < . Now, choose N ∈ N large enough for all n > N, |tn − t0| < δ. Then we ﬁnd for n > N

fn(t0) = fn(t0) − fn(tn) + fn(tn)

≤ fn(t0) − fn(tn)

≤ |fn(t0) − fn(tn)| < .

So the result holds as claimed.

Lemma 6.13. For all but ﬁnitely many n, cos(βn(t)) ≥ 0 for all t ∈ [0, 1].

Proof. We can interpret this as saying that the set

[ iβn(t) Y = {n ∈ N;

{nm} ⊂ N and for each m a tm ∈ [0, 1] such that cos(βnm (tm)) < 0. By compactness of [0, 1], we may pass to a subsequence (without relabelling, for ease of reading) such that {tm} converges to a limit, t0 ∈ [0, 1] say. The family of functions {fm = cos ◦βnm } is equicontinuous on D, by Lemma 6.11 and continuity of x 7→ cos(x). Restricting to t ∈ [0, 1], we maintain equicontinuity and also gain realness of the range of βn(t) (since for t ∈ [0, 1], U(t) is unitary). Hence we can 1 apply Lemma 6.12 to see that cos(βnm (t0)) < 2 for inﬁnitely many m ∈ N. But since U(t) − I is iβn(t ) compact, cos(βn(t0)) = < e 0 → 1 as n → ∞. This is a contradiction, so Y must have a ﬁnite number of elements. Hence the lemma is proven. −1 64 CHAPTER 6. PROPERTIES OF CV (K)C0(K)

We can now prove the hard part of Theorem 6.9.

iβn(1) Proof of the second claim of Theorem 6.9. Suppose we have some n with = e = sin(βn(1)) > ˙ 0, so βn(1) ∈ (2mπ, (2m + 1)π) for some m ∈ Z. Since βn(0) = 0 and βn(t) ≤ 0 for t ∈ [0, 1] by Lemma 6.10, βn(t) is strictly decreasing and hence m ≤ −1. Then by the intermediate value theorem, there exists a t0 ∈ [0, 1] with βn(t0) = −π. But then cos(βn(t0)) = −1 < 0. Applying Lemma 6.13 shows that this can only be satisﬁed by ﬁnitely many n. So the theorem is proven.

6.3.2 Flow of Eigenvalues

Throughout this calculation, subscripts of 0 will be used to denote objects related to the free problem, and subscripts of V will be used for the perturbed problem. When no subscript is used, ∞ the calculations will be relevant to both the free and perturbed case. Throughout, let V ∈ Cc (D) be non-negative and bounded away from the value 1. An important property of the scattering matrix used in the proof of the main result of [EP95] was that eigenvalues move monotonically around the circle for increasing k, so we would like a −1 similar result for the transmission operator CV (k)C0(k). It has been seen in numerical examples −1 1 that eigenvalues of CV (k)C0(k) do not move monotonically over the whole of S , with some eigenvalue paths turning back on themselves. See Section 5.3 for this numerical example. For ∞ this reason we only aim to show that under some conditions on V ∈ Cc non-negative, we can 1 −1 obtain some region around 1 in S where eigenvalues of CV (k)C0(k) move monotonically on the approach to 1 as k increases over some interval. Most of this calculation is valid on D as deﬁned at the beginning of this Chapter, although at some point we will have to pass to domain BR a ball of radius R > 0 centred at the origin and containing the support of V , to allow the computation to be continued. Theorem 6.6 shows that this reduction in generality does not do too much harm. We will split this calculation into a few propositions. iθ(k) −1 2 Suppose we have an eigenvalue e of CV (k)C0(k) with eigenfunction f(k) ∈ L (∂D) with ||f(k)||L2(∂D) = 1. By Corollary 6.3, these can be chosen to depend holomorphically on k providing we restrict to looking at some neighbourhood of S1 where the eigenvalues are discrete (i.e away 1 from 1). Due to Theorem 6.4, there exists some functions uA = uV , u0 in H (D) and smooth in the interior with 2 2 (−∆ + k A − k )uA = 0 on D 1 (6.6) f = k dnuA − iuA|∂D on ∂D and such that iθ Λ0u0|∂D + iu0|∂D = e (ΛV uV |∂D + iuV |∂D) . (6.7)

˙ R 2 2 2 R Proposition 6.14. θ = 4 D (V − 1)|uV | + |u0| + k2 ∂D (dnu0u0 − dnuV uV ) .

Proof. We have

−1 iθ(k) CV (k)C0(k)f(k) = e f(k) iθ(k) C0(k)f(k) = e CV (k)f(k). (6.8) 6.3. SPECTRAL PROPERTIES 65

Diﬀerentiating both sides with respect to k gives (using overdots to represent diﬀerentiation in k, and dropping ks from the notation)

˙ ˙ iθ iθ iθ ˙ C˙0f + C0f = iθe CV f + e C˙V f + e CV f.

iθ Taking the inner product with C0f = e CV f and using the fact that C0, CV are unitary, ||f|| = 1 and θ, θ˙ ∈ R gives

˙ ˙ iθ iθ iθ iθ iθ ˙ iθ C˙0f, C0f + C0f, C0f = iθe CV f, e CV f + e C˙V f, e CV f + e CV f, e CV f −1 ˙ ˙ ˙ iθ iθ −1 ˙ ˙ C0 C0f, f + f, f = iθ e CV f, e CV f + CV CV f, f + f, f −1 ˙ ˙ −1 ˙ C0 C0f, f = iθ + CV CV f, f ˙ −1 ˙ −1 ˙ iθ = C0 C0f, f − CV CV f, f . (6.9)

2 2 2 Now, for the moment suppose 0 6∈ σ(−∆D + k V − k ) ∪ σ(−∆D − k ). Then by deﬁnition C = (Λ + i)(Λ − i)−1 since both factors exist, so

C˙ = Λ˙ (Λ − i)−1 − (Λ + i)(Λ − i)−1Λ˙ (Λ − i)−1 and C−1C˙ = (Λ − i)(Λ + i)−1 Λ˙ (Λ − i)−1 − (Λ + i)(Λ − i)−1Λ˙ (Λ − i)−1

= (Λ − i)(Λ + i)−1Λ˙ (Λ − i)−1 − Λ˙ (Λ − i)−1 = (Λ − i)(Λ + i)−1Λ˙ (Λ − i)−1 − (Λ + i)(Λ + i)−1Λ˙ (Λ − i)−1 = −2i(Λ + i)−1Λ˙ (Λ − i)−1.

Substituting this into Equation 6.9 and simplifying gives

˙ −1 −1 −1 −1 θ = 2 (ΛV + i) Λ˙V (ΛV − i) f, f − (Λ0 + i) Λ˙0(Λ0 − i) f, f . (6.10)

Now, since Λ is self adjoint, we have by the functional calculus (Theorem 1.18)

∗ (Λ + i)−1 = (Λ∗ − i)−1 = (Λ − i)−1, so (Λ + i)−1Λ˙ (Λ − i)−1f, f = Λ˙ (Λ − i)−1f, (Λ − i)−1f . (6.11)

Remark 6.15. By holomorphicity of the extension of C to all k in some neighbourhood of the real axis, and the existence of (Λ − i)−1 on some neighbourhood of the real axis by Theorem 6.3 and its corollaries, the above formula works for all k ∈ R rather than just those for which 2 2 2 2 2 2 0 6∈ σ(−∆B +k V −k )∪σ(−∆B −k ). We will still require 0 6∈ σ(−∆B +k V −k )∪σ(−∆B −k ) for the time being so as to work with Λ˙ , but after doing so we can pass to arbitrary k again. −1 66 CHAPTER 6. PROPERTIES OF CV (K)C0(K)

Now, let’s try to determine a simplified form of Equation 6.10. We wish to show that θ˙ has fixed sign, at least close to θ = 1. 2 2 2 Since we are assuming 0 6∈ σ(−∆B + k V − k ) ∪ σ(−∆B − k ), we have that Λ0, ΛV exist. 1 Moreover ΛAuA|∂D = k dnuA for A = 0,V . To simplify notation, define gA := uA|∂D for A = 0,V . ˙ 1 1 Differentiating Equation 6.6 in k, we find that ΛAgA = k dnu˙A − k2 dnuA, whereu ˙A satisfies

d 2 2 2 2 0 = dk (−∆ + k A − k )uA = (−∆ + k A − k )u ˙A + 2k(A − 1)uA on D u˙A = 0 on ∂D.

The reason we hold gA ﬁxed in this diﬀerentiation is because we are looking at Λ˙AgA, and not ˙ (ΛAgA). See Section 1.2 for details. Since we are assuming A = 0,V to be real, u˙A will satisfy

(−∆ + k2A − k2)u˙ = −2k(A − 1)u on D A A , (6.12) u˙A = 0 on ∂D which will be necessary later. Combining these with Equations 6.10 and 6.11, we ﬁnd

−1 −1 Λ˙A(ΛA − i) f, (ΛA − i) f = Λ˙AgA, gA 1 1 = (d u˙ , u | ) − (d u , u | ) k n A A ∂BR k2 n A A ∂BR 1 Z 1 Z = dnu˙AuA − 2 dnuAuA. k ∂D k ∂BR

1 R Now, we can use the fact thatu ˙A = 0 on ∂D to subtract 0 = k ∂D u˙AdnuA and apply Green’s Formula (Theorem 2.1) to ﬁnd Z Z ˙ −1 −1 1 1 ΛA(ΛA − i) f, (ΛA − i) f = (dnu˙AuA − u˙AdnuA) − 2 dnuAuA k ∂D k ∂D 1 Z 1 Z = (uA∆u ˙A − u˙A∆uA) − 2 dnuAuA. k D k ∂D

Adding and subtracting terms, and using the fact that uA and u˙A satisfy the PDE of Equations 6.6 and 6.12, we ﬁnd Z Z ˙ −1 −1 1 1 ΛA(ΛA − i) f, (ΛA − i) f = (uA∆u ˙A − u˙A∆uA) − 2 dnuAuA k D k ∂D Z 1 2 2 2 2 = −uA(−∆ + k A − k )u ˙A +u ˙A(−∆ + k A − k )uA k D 1 Z − 2 dnuAuA k ∂D 1 Z 1 Z = (uA2k(A − 1)uA) − 2 dnuAuA k D k ∂D Z Z 2 1 = 2 (A − 1)|uA| − 2 dnuAuA. D k ∂D 6.3. SPECTRAL PROPERTIES 67

Combining this result with Equation 6.10, we have:

Z Z ˙ 2 2 1 θ = 2 2 (V − 1)|uV | + |u0| + 2 (dnu0u0 − dnuV uV ) . (6.13) D k ∂D

Simplifying gives the proposed result.

Remark 6.16. Note that this expression no longer requires existence of the Dirichlet to Neumann operators, so the assumption of Remark 6.15 is no longer necessary (but will be held in place for the time being for further simpliﬁcation later on). This point of the calculation will be referenced later on.

This looks like quite a nice formula for θ˙, although it is diﬃcult to see any reason why this should have a constant sign for θ near 1. To get some monotonicity, we can try using the fact (a simple calculation) that

2 −∆ − k , x∂x = −2∆,

Pn where [A, B] = AB − BA is the commutator of operators, and x∂x = i=1 xi∂xi is the generator of the dilation semigroup. In general, there are some major complications with considering the commutator of unbounded operators (see [RS80] for example), but since we are only applying this to u0 and uV which are smooth in the interior we will have no issues. Recalling that uA satisﬁes 2 2 (−∆ + k A − k )uA = 0 on BR (as does uA, since A is real), we then have

Z Z 2 2 (A − 1)|uA| = 2 uA(A − 1)uA D D 2 Z = 2 uA∆uA k D Z 1 2 = − 2 uA −∆ − k , x∂x uA k D Z Z 1 2 1 2 = − 2 uA(−∆ − k )(x∂xuA) + 2 uAx∂x((−∆ − k )uA) k D k D Z Z 1 2 = − 2 uA(−∆ − k )(x∂xuA) − uAx∂x(AuA) k D D 1 Z Z Z = − 2 uA(−∆)(x∂xuA) + uAx∂xuA − uAx∂x(AuA). k D D D −1 68 CHAPTER 6. PROPERTIES OF CV (K)C0(K)

Now, we can use Theorem 2.1 on the ﬁrst term, to obtain:

Z Z Z 2 1 1 2 (A − 1)|uA| = − 2 (−∆)uA(x∂xuA) − 2 [(dnuA)(x∂xuA) − uAdn(x∂xuA)] D k D k ∂D Z Z − uAx∂xuA − uAx∂x(AuA) D D Z Z 1 2 1 = − 2 (−∆ − k )uA(x∂xuA) − 2 [(dnuA)(x∂xuA) − uAdn(x∂xuA)] k D k ∂D Z − uAx∂x(AuA) D Z 1 Z = AuA(x∂xuA) − 2 [(dnuA)(x∂xuA) − uAdn(x∂xuA)] D k ∂D Z − uAx∂x(AuA). D

It is easy to check that x∂x is a derivation, i.e. x∂x(fg) = fx∂xg + gx∂xf. Using this above, we ﬁnd

Z Z Z 2 1 2 (A − 1)|uA| = AuA(x∂xuA) − 2 [(dnuA)(x∂xuA) − uAdn(x∂xuA)] D D k ∂D Z Z − uAuA(x∂xA) − uAA(x∂xuA) D D Z Z 2 1 = − |uA| (x∂xA) − 2 [(dnuA)(x∂xuA) − uAdn(x∂xuA)] . D k ∂D 6.3. SPECTRAL PROPERTIES 69

Substituting this back into Equation 6.13, we ﬁnd:

Z Z ˙ 2 2 1 θ = 2 2 (V − 1)|uV | + |u0| + 2 (dnu0u0 − dnuV uV ) D k ∂D Z Z 2 1 = 2 − |uV | (x∂xV ) − 2 [(dnuV )(x∂xuV ) − uV dn(x∂xuV )] D k ∂D Z Z 2 1 − − |u0| (x∂x0) − 2 [(dnu0)(x∂xu0) − u0dn(x∂xu0)] D k ∂D 1 Z + 2 (dnu0u0 − dnuV uV ) k ∂D Z Z 2 1 = 2 − |uV | (x∂xV ) − 2 [(dnuV )(x∂xuV ) − uV dn(x∂xuV )] D k ∂D 1 Z + 2 [(dnu0)(x∂xu0) − u0dn(x∂xu0)] k ∂D 1 Z + 2 (dnu0u0 − dnuV uV ) k ∂D Z 2 = −2 |uV | (x∂xV ) D 2 Z + 2 [(dnu0)(x∂xu0) − (dnuV )(x∂xuV )] k ∂D 2 Z + 2 [uV dn(x∂xuV ) − u0dn(x∂xu0)] k ∂D 2 Z + 2 [dnu0u0 − dnuV uV ] . k ∂D

We will progress to another proposition, with the assumption that D = BR is a ball centred at the origin. This simple geometry allows for signiﬁcant cancellations.

Proposition 6.17. Assume D = BR. Then

Z ˙ 2 θ = 2 |uV | (x∂xV ) + A sin θ + B(cos θ − 1), (6.14) BR where A and B can be given explicitly.

Proof. We will work in spherical polar coordinates, to utilise the fact that BR is a ball centred at the origin. In spherical polar coordinates, x∂x = r∂r, where r = |x| is the radial coordinate. We also ﬁnd that (∂ru) |r=R = dnu. So, we have

2 2 dn(r∂ru) = ∂r(r∂ru)|r=R = r∂r u|r=R + ∂ru|r=R = R∂r u|r=R + dnu. −1 70 CHAPTER 6. PROPERTIES OF CV (K)C0(K)

Using this fact in the previous formula, we ﬁnd: Z ˙ 2 θ = −2 |uV | (x∂xV ) BR 2 Z + 2 [(dnu0)(x∂xu0) − (dnuV )(x∂xuV )] k ∂BR 2 Z + 2 [uV dn(x∂xuV ) − u0dn(x∂xu0)] k ∂BR 2 Z + 2 [dnu0u0 − dnuV uV ] k ∂BR Z 2 = −2 |uV | (x∂xV ) BR 2 Z + 2 [(dnu0)(x∂xu0) − (dnuV )(x∂xuV )] k ∂BR 2 Z + 2 [uV dnuV − u0dnu0] k ∂BR Z 2R 2 2 + 2 uV ∂r uV |r=R − u0∂r u0|r=R k ∂BR 2 Z + 2 [dnu0u0 − dnuV uV ] . k ∂BR

Let’s examine the integrand of the boundary integral, which we will denote by I. Recalling that

(∂ru) |r=R = dnu, this integrand can be written as

2 2 I = (dnu0)(x∂xu0) − (dnuV )(x∂xuV ) + R uV ∂r uV |r=R − u0∂r u0|r=R 2 2 2 2 = R |dnu0| − |dnuV | + uV ∂r uV |r=R − u0∂r u0|r=R . (6.16)

Let’s try to deal with the second radial derivative terms. From Equation 2.3, the Laplacian can be written as n − 1 1 ∆u = ∂2u + ∂ u + ∆ u, r r r r2 tan 6.3. SPECTRAL PROPERTIES 71

n−1 where ∆tan is the standard Laplacian on S . Recalling that u0, uV satisfy Equation 6.6, we have for A = 0,V

n − 1 1 0 = −∆u + k2Au − k2u = −∂2u − ∂ u − ∆ u + k2Au − k2u . A A A r A r r A r2 tan A A A

Restricting to r = R, and noting that A|r=R ≡ 0 by deﬁnition of BR, we ﬁnd

n − 1 1 0 = −∂2u | − d u | − ∆ u | − k2u , r A r=R R n A r=R R2 tan A r=R A and so n − 1 1 u ∂2u | = − u d u | − u ∆ u | − k2|u |2, A r A r=R R A n A r=R R2 A tan A r=R A Substituting this into Equation 6.16, we ﬁnd

2 2 2 2 2 I = R |dnu0| − |dnuV | − k |uV | − |u0|

+ (n − 1) (u0dnu0|r=R − uV dnuV |r=R) 1 + (u ∆ u | − u ∆ u | ) . (6.17) R 0 tan 0 r=R V tan V r=R We shall see that the ﬁrst line of the previous formula for I cancels out. Recall Equations 6.6 and 6.7, which state:

iθ Λ0g0 − ig0 = ΛV gV − igV , Λ0g0 + ig0 = e (ΛV gV + igV ). (6.18)

Taking modulus squared of both sides of the ﬁrst equation (pointwise) gives

2 2 2 2 2 2 |Λ0g0 − ig0| = |Λ0g0| + |g0| + 2i=(g0Λ0g0) = |ΛV gV | + |gV | + 2i=(gV ΛV gV ) = |ΛV gV − igV | .

Similarly, taking modulus squared of both sides of the second equation gives

2 2 2 2 2 2 |Λ0g0 + ig0| = |Λ0g0| + |g0| − 2i=(g0Λ0g0) = |ΛV gV | + |gV | − 2i=(gV ΛV gV ) = |ΛV gV + igV | .

Adding these together, we ﬁnd that

2 2 2 2 |Λ0g0| + |g0| = |ΛV gV | + |gV | .

1 Then using the fact that ΛAgA = k dnuA and uA|∂BR = gA, we ﬁnd that on ∂BR, 1 1 |d u |2 + |u |2 = |d u |2 + |u |2. k2 n 0 0 k2 n V V Substituting this equality into Equation 6.17 shows that the ﬁrst line cancels out identically on

∂BR, so

I = (n − 1) (u0dnu0|r=R − uV dnuV |r=R) 1 + (u ∆ u | − u ∆ u | ) . (6.19) R 0 tan 0 r=R V tan V r=R −1 72 CHAPTER 6. PROPERTIES OF CV (K)C0(K)

We can take this computation a tiny bit further to obtain more explicit dependence on θ. Using Equation 6.18, and a linear equation solver, it is possible (read: easy, but messy) to show that

Similarly, we can use Equation 6.18 and the fact that dn and ∆tan commute to ﬁnd

(u0∆tanu0|r=R − uV ∆tanuV |r=R) 1 1 = (cos θ − 1) d u d (∆ u ) − u | ∆ u 2 k2 n V n tan V V ∂BR tan V sin θ − (u d (∆ u ) + d u ∆ u ) . (6.21) k V n tan V n V tan V

The form of Equation 6.22 and the expressions for A and B suggest we may obtain useful results if we restrict to potentials x∂xV of one sign. Since V is non-negative and tends to zero as r gets large, the only plausible assumption we could make on the sign of x∂xV is that x∂xV ≤ 0. Providing the interior integral and the terms A and B can be controlled independently of k in some region of R we will be able to get a ﬁxed sign on θ˙, since sin θ and cos θ − 1 both tend to zero as θ → 1. We will return dependence on k to each of our variables, as this now plays a signiﬁcant R 2 role. First of all, we shall bound |uV (k)| (x∂xV ) above, locally in k. For this we employ the BR following theorem.

Proposition 6.18. Assume that x∂xV ≤ 0, with equality not holding identically. For any bounded interval I ⊂ (0, ∞) there exists > 0 such that for all k ∈ I, Z 2 |uV (k)| (x∂xV ) < −. BR

Proof. Recall the following theorem. 6.3. SPECTRAL PROPERTIES 73

Theorem 6.19 (Carleman Estimate, [Zwo12], Theorem 7.7). Suppose U ⊂ D is open with compact closure in D, and let ﬁx k0 < k1 ∈ R. Then there exists constant C > 0 such that if u solves for some k0 < k < k1 the equation (−∆ + k2V − k2)u = 0 in D, then −Ck ||u||L2(U) ≥ e ||u||L2(D).

1 n This was proven in [Zwo12] for h = k and for D = R with V growing suﬃciently fast as x → ∞. However, reducing from the proof given in [Zwo12] to the case above is quite simple. To apply Theorem 6.19 to Equation 6.22, we ﬁrst take 0 > 0 small enough that the set U = {x ∈ 0 ∞ n D; x∂xV (x) < − } has positive measure (U will be an open set, as x∂xV (x) ∈ Cc (R )). Applying Theorem 6.19 provides a C > 0, such that Z Z 2 2 |uV | (x∂xV ) ≤ |uV | (x∂xV ) BR U 0 2 ≤ − ||uV ||L2(U) 0 −Ck 2 ≤ − e ||u || 2 . V L (BR) (6.23)

2 We will now show that ||uV || 2 can be uniformly bounded above and below locally in k. L (BR) Note by Theorem 6.3, CV (k) is a bounded operator (in fact unitary). From Equations 6.6 and Theorem 6.3, we ﬁnd

1 k u |∂B = (C (k)f(k) − f(k)), d u = (C f(k) + f(k)), V R 2i V n V 2 V which are hence also both locally uniformly bounded above and below in k, since ||f|| = 1. It is a theorem in [CK13] that the mapping from the Cauchy data {uV (k)|∂D, dnuV (k)} to uV (k) the solution of 2 2 (−∆ + k V − k )uV (k) = 0 is a uniformly bounded linear map which is also uniformly bounded below, locally in k. Hence 2 ||uV || 2 are uniformly bounded above and below locally in k as claimed. Used in Equation L (BR) R 2 6.23, we ﬁnd that |uV (k)| (x∂xV ) can be bounded above by − for some suﬃciently small BR > 0, locally in k.

A Black Box

We have just shown that the interior integral term in Equation 6.22 can be controlled. Dealing with A and B is more troublesome. Due to time constraints, we can only give a heuristic argument as to why A and B can be bounded locally in k for θ near 0. Our discussion from this point will be very un-rigorous, as making the following statements precise requires topics not covered in this thesis (such as microlocal analysis). −1 74 CHAPTER 6. PROPERTIES OF CV (K)C0(K)

−1 The idea is that the eigenfunctions of CV (k)C0(k) with eigenvalues approaching 1 from the upper half plane have frequencies bounded roughly by k. As A and B are essentially inner products of uA|∂BR , dnuA and ∆tanuA, frequency bounds on uA|∂BR should allow us to bound A and B.

But why should we expect frequency bounds on uA|∂BR ? Roughly, it is because uA|∂BR came −1 from an eigenvalue of CV (k)C0(k) which is far away from the accumulation at 1 (since the accumulation at 1 only occurs from below, by Theorem 6.9). To see why this makes sense, consider any operator with an accumulation of eigenvalues (at either some finite point, or ∞). Then eigenfunctions with eigenvalue close to the accumulation must become increasingly irregular as the eigenvalue approaches the accumulation point. Otherwise one could account for all the eigenvalues corresponding to “regular enough” eigenfunctions, and be left with only finitely many eigenvalues to share amongst the irregular functions. But regularity is in some sense finite, while irregularity can always be increased.

To quantify this in some sense, we could look at eigenfunctions of −∆[0,π]. This operator has spectrum {1, 4, . . . , n2,...}, with corresponding eigenfunctions x 7→ sin nx. The only accumulation point of the spectrum of ∆[0,π] is at ∞, and as n → ∞ eigenfunctions become more erratic. So if we restrict to looking at suﬃciently regular functions, ∆[0,π] will be “almost bounded”. For a more explicit discussion of this idea see [BH14] Appendix C, in which it is shown eigenfunctions of the Dirichlet to Neumann operator at frequency k are frequency bound on the boundary by 2k.

If we assume that the explanation in the Black Box can be made rigorous, Propositions 6.17 and 6.18 show that θ˙ is negative for θ suﬃciently close to 0 and for k in a small enough interval, 2 2 2 avoiding k for which 0 ∈ σ(−∆B + k V − k ) ∪ σ(−∆B − k ). We can apply Remark 6.15 to drop 2 2 2 the condition 0 6∈ σ(−∆B + k V − k ) ∪ σ(−∆B − k ). This result may not be satisfying, but we may proceed regardless.

6.3.3 1 as Eigenvalue

We revoke the assumption that D is a ball from here on, unless otherwise stated. −1 Our next theorem allows us to freely pass between eigenvalues of CV (k)C0(k) approaching 1 −1 as k ↑ k0, and eigenvalues 1 of CV (k0)C0(k0). This is an incredibly important distinction between transmission operator and scattering matrix, and relies on the fact that the transmission operator only depends on the open bounded set D.

n ∞ Theorem 6.20. Let D ⊂ R be open and bounded with smooth domain, and assume V ∈ Cc (D) −1 is non-negative and bounded away from the value 1. Suppose as k ↑ k0 > 0, CV (k)C0(k) has iθ(k) ˙ −1 eigenvalue e → 1. Then assuming that θ(k) does not converge to zero as k ↑ k0, CV (k0)C0(k0) −1 has eigenvalue 1. If there are M eigenvalues of CV (k)C0(k) approaching 1 (including degeneracy), −1 then CV (k0)C0(k0) has eigenvalue 1 with degeneracy M. Remark 6.21. The crux of this proof is the application of compactness theorems relevant to Sobolev spaces of bounded domains. This fails for the scattering matrix however, because for the n scattering matrix we would need to extend solutions to the whole of R . The condition that θ˙(k) does not converge to zero as k ↑ k0 is required to show that certain solutions to the semiclassical 6.3. SPECTRAL PROPERTIES 75 problems on D are non-zero. It might be possible to avoid this requirement, but how to do so is not currently known. Providing the Black Box of Section 6.3.2 can be made rigorous, this condition may be satisﬁed for D = BR and x∂xV ≤ 0. Noting Theorem 6.6, we could then restrict to any domain with smooth boundary contained in BR and containing the support of V and maintain eigenvalue 1 of the transmission operator, of degeneracy at least M.

Proof. The proof of this theorem requires many common compactness arguments relating to Sobolev spaces.

By Corollary 6.3, and Theorems 6.4 and 6.9, θ(k) depends smoothly on k as k ↑ k0 and there 1 exists uV (k), u0(k) ∈ H (D) depending holomorphically on k and smooth in the interior satisfying

2 2 2 (−∆ + k V − k )uV (k) = (−∆ − k )u0(k) = 0 in D with 1 1 d u (k) − iu (k)| = d u (k) − iu (k)| k n 0 0 ∂D k n V V ∂D and 1 1 d u (k) + iu (k)| = eiθ(k) d u (k) + iu (k)| . k n 0 0 ∂D k n V V ∂D We normalise so that 1 1 || d u (k) − iu (k)| || = || d u (k) − iu (k)| || = 1. k n 0 0 ∂D k n V V ∂D

Now, for k at which ΛV (k) and Λ0(k) exist, we ﬁnd: 1 1 = || d u (k) − iu (k)| ||2 k n 0 0 ∂D 2 = ||Λ0(k)u0(k) − iu0(k)|∂D||

= (Λ0(k)u0(k)|∂D − iu0(k)|∂D, Λ0(k)u0(k)|∂D − iu0(k)|∂D)

= (Λ0(k)u0(k)|∂D, Λ0(k)u0(k)|∂D) + (u0(k)|∂D, u0(k)|∂D)

+ (−iu0(k)|∂D, Λ0(k)u0(k)|∂D) + (Λ0(k)u0(k)|∂D, −iu0(k)|∂D) 2 2 = ||Λ0(k)u0(k)|∂D|| + ||u0(k)|∂D||

− i (u0(k)|∂D, Λ0(k)u0(k)|∂D) + i (u0(k)|∂D, Λ0(k)u0(k)|∂D) 1 = || d u (k)||2 + ||u (k)| ||2, k n 0 0 ∂D by self adjointness of Λ0(k). Similarly for uV (k). So 1 1 ||d u (k)||2 + ||u (k)| ||2 = ||d u (k)||2 + ||u (k)| ||2 = 1. (6.24) k2 n 0 0 ∂D k2 n V V ∂D

For k at which ΛV (k) and Λ0(k) do not exist we obtain the same equality by continuity, as explained in the remark following Theorem 6.1. As we have k ↑ k0 < ∞, this shows that 2 dnu0(k), dnuV (k), u0(k)|∂D and uV (k)|∂D are all uniformly bounded in L (∂D).

It is a theorem in [CK13] that the mapping from the Cauchy data {u|∂D, dnu} to u a solution of a PDE of the form we are considering is a bounded linear map, and so uV k and u0(k) are both −1 76 CHAPTER 6. PROPERTIES OF CV (K)C0(K)

2 uniformly bounded in L (D) as k ↑ k0. I claim these sets are also both uniformly bounded in H1(D). We have by Green’s Formula (Theorem 2.1) and the Cauchy-Schwarz inequality Z Z Z 2 |DuV (k)| dx = − uV (k)∆uV (k)dx + uV (k)dnuV (k) D D ∂D Z Z ≤ | uV (k)∆uV (k)dx| + | uV (k)dnuV (k)| D ∂D Z 2 ≤ k | uV (k)(V − 1)uV (k)dx| + ||uV (k)|∂D||L2(∂D)||dnuV (k)||L2(∂D) D 2 2 ≤ Ck ||uV (k)||L2(D) + ||uV (k)|∂D||L2(∂D)||dnuV (k)||L2(∂D),

where C is some bound on |V − 1|. The above is uniformly bounded in the region we are interested 2 in since k < k0 and we have already seen that the relevant L norms on the right hand side are 1 bounded. So uV k and u0(k) are both uniformly bounded in H (D) as k ↑ k0. Hence by the remark following the Rellich-Kondrachov Theorem of [Eva10] (Theorem 1, Section 5.7), we may 1 extract a subsequence kn ↑ k0 such that uV (kn) and u0(kn) have weak limits say uV , u0 in H (D) 2 respectively, such that convergence occurs in norm in L (D). We aim to show that uV , u0 are non-zero and satisfy the requirements of Theorem 6.4 to produce eigenvalue 1 of the transmission operator.

By compactness of the trace, as in Theorem 2.9, we ﬁnd that uV (kn)|∂D → uV |∂D and 2 u0(kn)|∂D → u0|∂D in norm in L (∂D) (since compact operators map weakly convergent sequences ∞ to norm convergent sequences, see Theorem VI.11 of [RS80]). So we ﬁnd for all φ ∈ Cc (D) Z Z Z 2 2 2 uV (−∆ + k0V − k0)φ = − uV ∆φ + k0 uV (V − 1)φ D D D Z Z Z 2 = DuV · Dφ − uV dnφ + k0 uV (V − 1)φ D ∂D D Z Z = lim DuV (kn) · Dφ − uV (kn)dnφ kn→k0 D ∂D Z 2 + kn uV (kn)(V − 1)φ D Z 2 2 = lim uV (kn)(−∆ + knV − kn)φ kn→k0 D = 0.

(Note: D applied to functions is the gradient operator, not to be confused with the domain D). 1 We have been able to introduce the limits since uV (kn) → uV weakly in H (D), and uV (kn)|∂D → 2 uV |∂D in norm in L (∂D), and the ﬁnal equality due to the fact that uV (kn) is a strong solution 2 2 to (−∆ + knV − kn)uV (kn) = 0, and is hence a weak solution. Since φ was arbitrary, uV is a 2 2 weak solution to (−∆ + k0V − k0)uV = 0, and is hence a strong solution by elliptic regularity (see 2 [Eva10]). Similarly u0 is a strong solution to (−∆ − k0)u0 = 0. We still need to check that the boundary conditions of Theorem 6.4 are satisﬁed by uV and u0, and that these two functions do not vanish. 6.3. SPECTRAL PROPERTIES 77

2 Note that since dnuV (k) is uniformly bounded in L as k ↑ k0, there exists a weak limit of 2 2 {dnuV (kn)} in L (∂D) after possibly passing to a subsequence, since L spaces are reflexive and hence norm-closed balls are weakly compact (see [RS80] for details). I claim that this weak limit is the normal derivative of uV . By examining the definition of the normal derivative (Definition 2.8), we find that for all w ∈ H1(D)

Z Z Z Z 2 DuV · Dw + w∆uV = DuV · Dw + k w(V − 1)uV D D U U Z Z 2 = lim DuV (kn) · Dw + kn w(V − 1)uV (kn) kn→k U U Z Z = lim DuV (kn) · Dw + w∆uV (kn) kn→k U U Z = lim wdnuV (kn)dS kn→k ∂U

So dnuV (kn) → dnuV weakly as claimed. Similar working shows that the same is true of u0(kn) and u0. So 1 d u − iu | is the weak limit of 1 d u (k ) − iu (k )| , and 1 d u − iu | is the k0 n 0 0 ∂D kn n 0 n 0 n ∂D k0 n V V ∂D weak limit of 1 d u (k ) − iu (k )| . But by deﬁnition of u (k ) and u (k ), 1 d u (k ) − kn n V n V n ∂D V n 0 n kn n 0 n iu (k )| = 1 d u (k ) − iu (k )| for all n, so by uniqueness of weak limits 0 n ∂D kn n V n V n ∂D

1 1 dnu0 − iu0|∂D = dnuV − iuV |∂D. k0 k0

We can repeat the same working to ﬁnd

1 1 dnu0(k0) + iu0(k0)|∂D = dnuV (k0) + iuV (k0)|∂D, k0 k0

iθ(kn) noting e → 1 as kn ↑ k0.

So now it suﬃces to check that uV and u0 are not zero, in which case Theorem 6.4 with the −1 pair uV , u0 will produce the desired result that 1 is an eigenvalue of CD,V (k0)CD,0(k0). For this, we need the last assumption about the limit of θ˙ not being 0. The main issue with what we have so far is that the convergence of normal derivatives is only weak and so dnuV may be zero. In the 1 2 2 equation 2 ||dnuV (kn)|| + ||uV (kn)|∂D|| = 1 we may have increasingly large norms of the normal kn derivative term while the trace term tends to zero, only to have both vanish in the limit.

So, suppose that uV = 0. Recall Proposition 6.14, which states that

Z ˙ 2 2 θ(kn) = 4 (V − 1)|uV (kn)| + |u0(kn)| D 2 Z + 2 dnu0(kn)u0(kn) − dnuV (kn)uV (kn) . k ∂D

(Note that normalisation of Equation 6.24 is equivalent to normalisation of the f from the proof of

Proposition 6.14, so this formula is valid). By Equation 6.24, we ﬁnd that ||dnuV (k)||, ||dnu0(k)|| ≤ −1 78 CHAPTER 6. PROPERTIES OF CV (K)C0(K) k. Combining with the Cauchy-Schwarz inequality we ﬁnd Z Z ˙ 2 2 2 |θ(kn)| ≤ 4 |V − 1||uV (kn)| + |u0(kn)| + 2 dnu0(kn)u0(kn) D k ∂D Z 2 + 2 dnuV (kn)uV (kn) k ∂D 2 2 2 ≤ 4C||u (k )|| + 4||u (k )|| + ||d u (k )|| 2 ||u (k )| || 2 V n L2(D) 0 n L2(D) k2 n 0 n L (∂D) 0 n ∂D L (∂D) 2 + ||d u (k )|| 2 ||u (k )| || 2 k2 n V n L (∂D) V n ∂D L (∂D) 2 2 2 ≤ 4C||u (k )|| + 4||u (k )|| + ||u (k )| || 2 V n L2(D) 0 n L2(D) k 0 n ∂D L (∂D) 2 + ||u (k )| || 2 k V n ∂D L (∂D)

2 Where C is some bound on |V − 1|. If we suppose that uV , u0 are both zero, then L convergence of uV (kn) to uV both in the interior of D and on the boundary and positivity of k0 implies that all ˙ the terms above converge to 0 as kn ↑ k0 which contradicts our assumption on θ(k). Hence uV and u0 are not both zero, and satisfy the requirements of Theorem 6.4 to produce 1 as an eigenvalue of −1 the transmission operator CD,V (k)CD,0(k). Note that we used boundedness of normal derivative terms to avoid the issues of convergence of normal derivative terms being only in the weak sense.

In the case that there are M eigenvalues approaching 1 as k ↑ k0, we can repeat this same process with each eigenspace, obtaining M linearly independent pairs of uV , u0 to which Theorem −1 6.4 gives 1 as an eigenvalue of CD,V (k)CD,0(k) of degeneracy M.

Remark 6.22. Assuming the condition on θ˙ can be veriﬁed, Theorem 6.20 shows that Conjecture C cannot possibly hold in converse. If the converse to ConjectureC were to hold, then Theorem

6.20 says that whenever the transmission operator has eigenvalues approaching 1 as k ↑ k0, the scattering matrix must attain eigenvalue 1 at frequency k0. But as discussed in Section 4.2 there are many domains on which the scattering matrix cannot possibly take eigenvalue 1, while it seems generic that the transmission operator should have eigenvalues which approach 1.

Remark 6.23. We may also combine Theorem 6.20 with the working of Section 6.1, and the fact that the scattering matrix for central potentials has eigenvalues which can be tracked continuously through 1 (eigenfunctions are always the spherical harmonics of Section 2.1.1; an easy exercise in separation of variables) to see that ConjecturesB andC are in fact equivalent for central potentials. That is, eigenvalues of the transmission operator approach 1, if and only if eigenvalue 1 is attained, if and only if scattering matrix attains eigenvalue 1, if and only if scattering matrix has eigenvalues approaching 1. Since we have already seen that ConjectureC holds for central potentials, verifying the constraint on θ˙ of Theorem 6.20 would instantly prove ConjectureB for central potentials. Bibliography

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