On Spectral Inequalities in Quantum Mechanics and Conformal Field Theory

DEGREE PROJECT, IN MATHEMATICS , SECOND LEVEL STOCKHOLM, SWEDEN 2015

OSCAR MICKELIN

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ENGINEERING SCIENCES

On Spectral Inequalities in Quantum Mechanics and Conformal Field Theory

O SCAR M ICKELIN

Master’s Thesis in Mathematics (30 ECTS credits) Master Programme in Mathematics (120 credits) Royal Institute of Technology year 2015 Supervisor at KTH was Ari Laptev Examiner was Ari Laptev

TRITA-MAT-E 2015:17 ISRN-KTH/MAT/E--15/17--SE

Royal Institute of Technology School of Engineering Sciences

KTH SCI SE-100 44 Stockholm, Sweden

URL: www.kth.se/sci

iii

Abstract

Following Exner et al. (Commun. Math. Phys. 26 (2014), no. 2, 531541), we prove new Lieb-Thirring inequalities for a general class of self-adjoint, second order dierential operators with matrix-valued potentials, acting in one space- dimension. This class contains, but is not restricted to, the magnetic and non-magnetic Schrödinger operators. We consider the three cases of functions dened on all reals, all positive reals, and an interval, respectively, and acquire three dierent kinds of bounds. We also investigate the spectral properties of a family of operators from conformal eld theory, by proving an asymptotic phase-space bound on the eigenvalue counting function and establishing a number of spectral inequalities. These bound the Riesz-means of eigenvalues for these operators, together with each individual eigenvalue, and are applied to a few physically interesting examples.

Keywords: Schrödinger operators, Lieb-Thirring inequalities, commutation method, conformal eld theory, Birman-Schwinger principle. iv

Sammanfattning

Vi följer Exner et al. (Commun. Math. Phys. 26 (2014), nr. 2, 531541) och bevisar nya Lieb-Thirring-olikheter för generella, andra gradens självadjunger- ade dierentialoperatorer med matrisvärda potentialfunktioner, verkandes i en rumsdimension. Dessa innefattar och generaliserar de magnetiska och icke- magnetiska Schrödingeroperatorerna. Vi betraktar tre olika fall, med funktion- er denierade på hela reella axeln, på den positiva reella axeln, samt på ett interval. Detta resulterar i tre sorters olikheter. Vidare undersöker vi spektralegenskaperna för en klass operatorer från konform fältteori, genom att asymptotiskt begränsa antalet egenvärden med ett fasrymdsuttryck, samt genom att bevisa ett antal spektralolikheter. Dessa be- gränsar Riesz-medelvärdena för operatorerna, samt varje enskilt egenvärde, och tillämpas på ett par fysikaliskt intressanta exempel.

Nyckelord: Schrödingeroperatorer, Lieb-Thirring-olikheter, kommutationsme- toden, konform fältteori, Birman-Schwinger-principen.

Acknowledgements

I am indebted, and would like to express my sincere gratitude, to my supervisor Ari Laptev, for introducing me to the use of spectral theory in mathematical physics. By suggesting more interesting problems than those that could be approached in this thesis, and by continually providing support and encouragement along the way, he has been a great inspiration. For all of your time spent, all theorems and approaches you have taught me, and all of our chats: thank you! Since this thesis marks the end of my studies at KTH, I would also like to thank all of the people I have gotten to know. As usual, listing all the names would not be possible on a single page, so is therefore left as an exercise to the reader. A last expression of gratitude is due to the many professors who have fueled my interest in nding things out, and then especially to those who responded to my many questions why is X?, with a smile and because Y.

Oscar Mickelin Stockholm, May 2015

Contents

Acknowledgements v

Contents vi

List of Figures vii

1 Introduction 1 1.1 Motivation ...... 1 1.2 Organization ...... 2

2 Preliminaries 3 2.1 Notation, conventions and denitions ...... 3 2.2 Useful theorems ...... 6

3 Lieb-Thirring Inequalities 9 3.1 The stability of matter problem ...... 9 3.2 Problem statement ...... 13 3.3 Summary of results ...... 14 3.4 Examples, remarks and comments ...... 16

4 Operators from Conformal Field Theory 21 4.1 Introduction and problem statement ...... 21 4.2 Example: spectral properties of Hµ ...... 23 4.3 Eigenvalue estimates for general potentials ...... 34

Bibliography 51

Paper A 57

List of Figures

3.1 Numerical tightness-check for Lieb-Thirring inequalities...... 20 (a) Q as a function of M, γ...... 20 (b) Q for an asymptotic range of parameters...... 20 4.1 Eigenvalue-bounding functions in Theorem 14...... 44 (a) The right hand side of Equation (4.174) plotted for complex λ. . 44 (b) The right hand side of Equation (4.174) plotted for real λ. . . . . 44 4.2 Numerical calculations of constants in Corollary 6...... 48 (a) D(p, b)...... 48 (b) D(b) for |{x : 0 < Λ − V (x)}| = 1...... 48

vii

1 Introduction

1.1 Motivation

Many of the tools in functional analysis have been created in order to answer questions from physics. Harmonic analysis, variational calculus and the theory of partial dierential equations all originate from, and continue to contribute to, physical applications. In particular, purely mathematical work concerning the Schrödinger operator, which can be written as

− ∆ + V (x), (1.1) has led to advances in perturbation theory, operator theory and spectral theory, on the mathematical side of things. On the physical side, the theoretical work has also found immediate applications, as well as opened up new means of interpretation and methodology. There are therefore both mathematical and physical reasons to rigorously study questions arising in physics, as will be done in this thesis. One question which has received signicant attention during the latter half of the previous century concerns the stability of matter. The mechanisms underlying stability properties of both isolated atoms and bulk matter have been understood in a number of settings, but are neither physically nor mathematically obvious. The main tool used in proving that fermionic bulk mass is stable, is an inequality due to Lieb and Thirring [Lie76], which has since been found useful in both physics and mathematics. More results have been discovered after suitably generalizing this inequality, and the rst part of this thesis proves a number of new generalized Lieb-Thirring inequalities. Another area of active mathematical and physical research is conformal eld theory, which describes quantum eld theories invariant under conformal transformations [Blu09]. Examples of these include string theory and the Ising model, where a number of mathematically interesting operators are dened. The second part of this thesis starts with a detailed example concerning the spectrum of such an operator, and concludes by proving a number of spectral bounds for a family of operators.

1 2 CHAPTER 1. INTRODUCTION

These bounds are obtained by adapting arguments and techniques from classical spectral theory to this new setting, and can therefore also be seen as something of an introduction to many classical results.

1.2 Organization

The remainder of this text is organized as follows. Notation and denitions are rst listed in Chapter 2, together with a few useful theorems to be used in the remainder of the text. Chapter 3 then introduces the stability of matter problem and motivates the questions investigated in the rst part of the thesis. A summary of the results obtained is then provided in Section 3.3, with full details available in [Mic15]. In Chapter 4, we lastly investigate a family of operators from conformal eld theory. Motivated by the rst part of the thesis, we begin with a detailed study of an example in Section 4.2, after which we prove a number of spectral inequalities in Section 4.3. 2 Preliminaries

2.1 Notation, conventions and denitions

We rst x the notation and conventions to be used in the text. All concepts in this section are standard and can be found in e.g. [Ree72, Ree78, Fri82, Maz85]. In what follows, the reader will be assumed familiar with basic quantum mechanics and spectral theory at the level of e.g. [Ree72, Tak08].

Sets and symbols

We let R+ := [0, ∞) as a shorthand. The letter Ω will usually refer to a subset of n p×q R . For R a ring, we take R as the ring of all p × q-matrices with entries in R. When R denotes functions with values in R or C, integrability and dierentiability assumptions of elements in Rp×q will typically be understood element-wise. The restriction of an operator to a space will be denoted by , and the domain T X T X of T by D(T ). We will write |A| for the Lebesgue-measure of a set A, and χA for the characteristic function of A. For two normed spaces , , we will say that there is a continuous (X, k·kX ) (Y, k·kY ) inclusion from X into Y if there is a positive constant C with

(2.1) kψkY ≤ C kψkX , for all ψ ∈ X. We will denote this as X,→ Y .

Function spaces

Take n and as a normed space. For some with , we let Ω ⊆ R (X, k·kX ) p 1 ≤ p ≤ ∞ Lp(Ω; X) denote

( 1 ) Z p p p (2.2) L (Ω; X) := ψ :Ω → X : kψkLp(Ω;X) := kψ(x)kX dx < ∞ . Ω

3 4 CHAPTER 2. PRELIMINARIES

p When X = R with the usual norm, we will also simply write L (Ω). We will most m p×q often use X = C for some integer m and the natural norm. For the case X = C , the norm used in Equation (2.2) is the maximum norm of the entries (i.e. not k·kX the operator norm) to be able to denote element-wise integrability assumptions in a convenient manner. We denote the set of k times dierentiable functions with values in R or C on Ω by Ck(Ω), the set of smooth functions by C∞(Ω), and the set of smooth functions with compact support by ∞ . C0 (Ω) A (not necessarily continuous) function 0 n is a weak partial derivative fi : R → R n of a function f : R → R if Z Z 0 ∂φ(x) (2.3) fi (x)φ(x)dx = − f(x) dx, Rn Rn ∂xi for all ∞ n . Iterating this denition, we will use multi-index notation to φ ∈ C0 (R ) denote higher derivatives. Let |α| = k; the order α weak derivative of f will then be denoted by ∇αf. Using this, we let W k,p(Ω) denote the Sobolev space of functions n on the open set Ω ⊆ R , by dening

W k,p(Ω) := {ψ ∈ Lp(Ω) : ∇αf ∈ Lp(Ω), for all |α| ≤ k} . (2.4)

We recall that W k,p(Ω) is a Banach-space with the norm

X α (2.5) kψkW k,p(Ω) := k∇ ψkLp(Ω) . |α|≤k

The convenient notation Hk(Ω) := W k,2(Ω) will also be used in the following, and Hk(Ω) is a Hilbert-space with the inner product

X α α (2.6) hψ, φiHk(Ω) := h∇ ψ, ∇ φiL2(Ω). |α|≤k

n We will also refer to the Schwartz-space S(R ), consisting of smooth functions with derivatives decreasing faster than any polynomial. In symbols, this can be dened as

n ∞ n α β for all (2.7) S(R ) := ψ ∈ C (R ) : sup |x| · ∇ ψ(x) ≤ ∞, α, |β| ≥ 0 . x∈Rn Lastly, we will talk about Hölder spaces, or Hölder continuous functions, which consist of functions with strong regularity properties. Let l ∈ N be a positive integer, and α ∈ [0, 1]. We denote by Cl,α the space of functions ψ for which

∇βψ(x) − ∇βψ(y) (2.8) |ψ|Cl,α := sup α x6=y, |x − y| |β|=l 2.1. NOTATION, CONVENTIONS AND DEFINITIONS 5 is nite. We endow this space with the norm

β (2.9) kψkCl,α := sup ∇ ψ(x) + |ψ|Cl,α , n x∈R , |β|≤l which turns Cl,α into a Banach-space.

Operator spaces

We next dene two useful spaces of operators. Let (X, h·, ·iX ) be a separable Hilbert- space with basis ∞ . If is a bounded, linear operator on , we dene the {en}n=1 T X trace of T by ∞ X Tr(T ) := hT en, eniX . (2.10) n=1 This sum is independent of the choice of the basis ∞ ; if it is nite, is said {en}n=1 T to be of trace-class. We denote all such T by

I1(X) := {T : X → X : Tr(T ) < ∞} , (2.11) with the associated and actually well-dened norm

kT k := Tr (|T |) . (2.12) I1(X) We use this, in turn, to dene the class of Hilbert-Schmidt operators. Take

∗ I2(X) := {T : X → X : Tr(T T ) < ∞} , (2.13) and denote by kT k the natural norm I2(X)

kT k := pTr(T ∗T ). (2.14) I2(X) These two classes of functions are related, as seen from the following result.

Theorem 1 ([Ree72]). The following properties hold.

1. S is in I1(X) if and only if S = T1T2, for T1,T2 ∈ I2(X).

2. kSk ≤ kSk ≤ kSk . X I2(X) I1(X)

3. Every S in I1(X) or I2(X) is compact.

Out of I1(X), I2(X), the latter will be of more use in the following, since its norm can be explicitly computed in most situations of interest. We have the following result. 6 CHAPTER 2. PRELIMINARIES

2 Theorem 2 ([Ree72]). An operator T is Hilbert-Schmidt in L (R) if and only if there exists some function K(x, y) with Z (T ψ)(x) = K(x, y)ψ(y)dy, (2.15) R and Z |K(x, y)|2 dxdy < ∞. (2.16) R2 In this case, we have Z kT k2 = |K(x, y)|2 dxdy. (2.17) I2(X) R2 2.2 Useful theorems

This section lists some theorems in forms which will be useful at some point in the text. They are stated mostly without proof, and in less general form than is possible to achieve, for the sake of accessibility. This is intended to provide a quick reference for the reader, in a form streamlined for the exposition. Full proofs, statements, and longer discussions of the results can be found where indicated.

n Theorem 3 (Sobolev embedding theorem; [Bre83, Maz85, Str08]). Let Ω ⊆ R be an open ball of arbitrary radius. Depending on the relationship of the parameters k, p, n, we then have the following inclusions. 1. If , then k,p q , where np . kp < n W (Ω) ,→ L (Ω) 1 ≤ q ≤ n−pk 2. If kp = n, then W k,p(Ω) ,→ Lq(Ω) for all 1 ≤ q < ∞.

bk− n c, n −b n c 3. If , and n , then k,p p p p . kp > n k − p 6∈ N W (Ω) ,→ C (Ω)

bk− n c,α 4. If , and n , then k,p p , for any kp > n k − p ∈ N W (Ω) ,→ C (Ω) 0 ≤ α ≤ 1. Corollary 1 (Adapted from [Str08]). For n , we have the (not necessarily k > 2 + 2 continuous) inclusion k n 2 n H (R ) ⊆ C (R ). (2.18) k n Proof. Since any element of H (R ) is square-integrable, it will be enough to show that Hk(Ω) ,→ C2(Ω), (2.19) where Ω is an open ball of arbitrary radius. This follows from the third case in Theorem 3 above.

n Theorem 4 (Young's inequality; [Lie01c]). Take f, g, h : R → R, and let p, q, r ≥ 1 be real numbers with p−1 + q−1 + r−1 = 2. We then have the inequality Z f(x)g(x − y)h(y)dxdy ≤ C kfk p n kgk q n khk r n . (2.20) p,q,r;n L (R ) L (R ) L (R ) R2n 2.2. USEFUL THEOREMS 7

Here, the constant Cp,q,r;n can be written as

n Cp,q,r;n = (CpCqCr) , (2.21) where p1/p C2 = , (2.22) p p01/p0 0 −1 0−1 and p, p are Hölder conjugate through p + p = 1. Cq and Cr are dened analogously. The constant Cp,q,r;n is also sharp. Theorem 5 (Schur's lemma; [Gra09]). Consider the integral operator Z (TK ψ)(x) := K(x, y)ψ(y)dy. (2.23) R 2 A sucient condition for TK to be a bounded, linear operator on L (R) is that K satises the two conditions Z |K(x, y)| dx ≤ C a.e. in y, (2.24) R Z |K(x, y)| dy ≤ C a.e. in x, (2.25) R for some constant C. In this case, it also holds that

Z Z ! 2 kTK k ≤ sup |K(x, y)| dy · sup |K(x, y)| dx . (2.26) x∈R R y∈R R

3 Lieb-Thirring Inequalities

This chapter deals with the rst problem in the thesis, namely Lieb-Thirring inequalities, and their relation to the stability of matter. Serving as motivation for the problems in the thesis, the exposition in Section 3.1 largely follows those of [Lie01c, Lie09, Sei10], where more details can be found. We then dene a problem statement in Section 3.2, provide a solution in Section 3.3, and consider a few examples in Section 3.4.

3.1 The stability of matter problem

Stability of the rst kind One of the chronologically rst important results in quantum mechanics concerns why isolated atoms are stable and their electrons do not collapse into their nuclei. This empirical fact is termed stability of the rst kind and will now be explained. By several orders of magnitude, the largest force acting on an electron bound by an atom with atomic number Z, is the Coulomb interaction. For the sake of simplicity, we will therefore work in the non-relativistic limit, neglect other forces, e.g. gravity and magnetic forces, and write our Hamiltonian as 1 Zα H := − ∆ − , (3.1) 2 |x|

2 after choosing our units appropriately. Here, α = e ≈ 1 is the ne-structure ~c 137 3 1 3 constant, and x ∈ R . The total energy of a wave-function ψ ∈ H (R ) can then be written as Z 1 Zα E(ψ) := |∇ψ(x)|2 − |ψ(x)|2 dx. (3.2) R3 2 |x| which gives the ground-state energy

E0 := inf E(ψ). (3.3) ψ∈D(H), kψk 2 3 =1 L (R ) 9 10 CHAPTER 3. LIEB-THIRRING INEQUALITIES

Both E(ψ) and E0 are dened analogously when the Coulomb potential is replaced by a more general potential V . Physically, electrons described by the wave-function ψ will tend towards lower, more energetically favorable energy levels, and the dierence in energy when changing levels is emitted in the form of light. Stability of the rst kind is then equivalent to the emission of at most a nite energy, i.e. to imposing that E0 > −∞. This is the case for the Coulomb potential, and even for more general types of potentials. Indeed, one can use the Sobolev embedding theorem to show that if 3/2 3 ∞ 3 V ∈ L (R ) + L (R ) denotes a potential function, then the Hamiltonian 1 H := − ∆ + V (x) (3.4) V 2 induces stability of the rst kind. In fact, there exist strictly positive constants C and D such that

CE(ψ) ≥ k∇ψk 2 3 − D kψk 2 3 ≥ −D kψk 2 3 . (3.5) L (R ) L (R ) L (R ) See e.g. [Lie01c] for a derivation.

Stability of the second kind Another more recently considered stability problem arises when modelling bulk matter. We will study a system consisting of N electrons orbiting M nuclei, with atomic number Zk, k = 1,...,M, respectively. The nuclei will be assumed xed at the positions Rk, and the positions of the electrons will be denoted by xi, i = 1,...,N. Neglecting magnetic and gravitational forces once again, we will study the Hamil- tonian 1 H := − ∆ + αV (x, R), (3.6) 2 where x = (x1, . . . , xN ), R = (R1,...,RM ) and the multi-body Coulomb potential V is given by

E0 := inf E(ψ). (3.8) 3M ψ∈D(H),R∈R , kψk 2 3N =1 L (R ) Removing the two positive terms in V , our potential is bounded below by sums of potentials of the form in the case of isolated atoms. Since all of these induce stability of the rst kind, it follows that also V does. The bulk matter does however introduce the following additional stability problem. In the thermodynamic limit of 3.1. THE STABILITY OF MATTER PROBLEM 11 large N and M, the total energy of the modelled system equals k · (N + M), where k = k(Z1,...,ZM ) is a constant. Relaxing this equality to an inequality, it follows that, at least in this limit, we certainly have the bound

E0 ≥ −k · (N + M). (3.9) A system for which Equation (3.9) holds for all N and M, and not just asymptotically, is said to be stable of the second kind [Fis66, Dys67, Lie09]. The rst point of this denition is that systems that do not have this extensive stability property would behave dierently from physical intuition. If, for instance, E0 would be pro- portional to a higher power of the particle numbers, then mixing two sets of particles would release a large amount of energy spontaneously. The second point of the denition is that not all matter is stable of the second kind. In fact, bosonic bulk matter asymptotically satises 5/3 E0 = −C min(N,M) , (3.10) for some constant C [Lie09]. This can be expected intuitively: the ground-state energy in Equation (3.10) means that it is energetically favorable for bosons to clump together, and more so than fermions. This is in line with an intuitive interpretation of the exclusion principle. Stability of the second kind is therefore intrinsically fermionic in nature, and, in particular, does not follow from stability of the rst kind. Understanding the mechanisms giving rise to stability of the second kind is therefore of physical interest, and we will see below that the proof is also mathematically interesting. We remark in passing that also matter with a relativistic Hamiltonian, even with magnetic elds present, can be shown to be stable. This is however only true if the ne-structure constant α is smaller than some critical value αc; fortunately, the numerical value of α does happen to be smaller than αc. See e.g. [Lie88, Lie97a, Lie97b, Fra07] and the references therein for more information. To conclude this section and to motivate the theme of the thesis, we now proceed to prove Equation (3.9). We will need the following three theorems, stated without proof. The last of these are the so-called Lieb-Thirring inequalities, which will be central in the remainder of this chapter.

Theorem 6 ([Lie09]). E0 decreases if all atomic numbers Zj for j = 1,...,M are replaced by some Z ≥ maxj(Zj). Theorem 7 (Baxter's inequality; [Lie09]). Let V (x, R) be a Coulomb potential with N electrons and M nuclei of equal atomic number Z, i.e. N M X 1 X X Z X Z2 V (x, R) = − + , (3.11) |xi − xj| |xi − Rk| |Rk − Rl| 1≤i

Theorem 8 (Lieb-Thirring inequalities; [Hun02, Lie76, Wei96, Cwi77, Lap97, Lie01b, 2 d Roz76]). Study L (R ) and take γ ∈ R+, with  γ ≥ 1 , d = 1,  2 γ > 0, d = 2, (3.13) γ ≥ 0, d ≥ 3.

Assume γ+d/2 1 d and let ∞ denote the non-positive eigenvalues of V− ∈ L (R ) {−Ek}k=1 H = −∆ + V (x). (3.14)

There then exist constants Lγ,d such that ∞ Z X γ γ+d/2 (3.15) |Ek| ≤ Lγ,d V (x)− dx. d k=1 R Using these results, we can prove stability of the second kind for the Coulomb potential, following [Sol06]. We study electrons, but might just as well allow for any fermions and therefore let our Hamiltonian act on anti-symmetric wavefunctions with q possible spin-states. By Theorem 6, we can assume that all nuclei have the same atomic number Z. Baxter's inequality now implies N 1 X 1 (2Z + 1)α − ∆ + αV ≥ − ∆ − , (3.16) 2 2 i min |x − R | i=1 j i j where ∆i is the Laplacian acting on the coordinates of the ith fermion. Fixing a λ ∈ R+, we will perform a trick and bound E0 by minimizing N X 1 (2Z + 1)α hHψ, ψi 2 3N ≥ − ∆ − ψ, ψ (3.17) L (R ) 2 i min |x − R | i=1 j i j L2(R3N ) N X 1 (2Z + 1)α = −λN + − ∆ + λ − ψ, ψ . 2 i min |x − R | i=1 j i j L2(R3N ) (3.18) Since we are dealing with fermions, at most q particles can occupy the same state, by the exclusion principle. Using the min-max principle, we can then bound the right hand side of Equation (3.18) from below using the sum of the N lowest eigenvalues of the isolated Hamiltonian 1 (2Z + 1)α − ∆ + λ − , (3.19) 2 minj |x − Rj| together with this spin constraint. Overcounting grossly, we instead sum over all the negative eigenvalues −λn ≤ 0 of the isolated Hamiltonian in Equation (3.19), and do this q times to obtain ∞ X (3.20) hHψ, ψiL2(R3N ) ≥ −λN − q |λn| . n=0 3.2. PROBLEM STATEMENT 13

Using the Lieb-Thirring inequality with γ = 1, d = 3 for the operator in Equa- tion (3.19), we can write

∞ 5 Z 2 X 1 5 (2Z + 1)α − |λn| ≥ − 2 2 L1,3 λ − dx (3.21) 2 3 min |x − R | n=0 R j j − 5 3 Z (2Z + 1)α 2 ≥ −2 2 L1,3M λ − dx (3.22) R3 |x| − 2 3 3 3 5π (2Z + 1) α = −2 2 L1,3 √ M. (3.23) 4 λ

3 The factor 2 in Equation (3.22) is inherited from the factor 1 in front of the 2 2 Laplacian in Equation (3.6), as compared to Equation (3.14). Inserting this into Equation (3.20) yields

2 3 3 3 5π (2Z + 1) α E0 ≥ −λN − q2 2 L1,3 √ M, (3.24) 4 λ which implies stability of the second kind using any positive λ. In order to achieve a tighter bound, we minimize the magnitude of this expression with respect to λ, which also allows us to compare our bound to empirical results. This gives

3 2 2 2 2 2 1 E ≥ − (5π L q) 3 (2Z + 1) α M 3 N 3 (3.25) 0 2 1,3 3 2 2 2 2 ≥ − (5π L q) 3 (2Z + 1) α (M + N) . (3.26) 2 1,3 As an example, we can mention neutral hydrogen, which has Z = 1, N = M, and q = 2. Inserting numerical values into Equation (3.26) yields the bound E0 ≥ −30.52·M Rydberg [Lie09], which has an order of magnitude consistent with empirical results. A more rened analysis can decrease the magnitude of the constant by roughly four times; see e.g. [Lie09] for details.

3.2 Problem statement

The main tools of the last section were the Lieb-Thirring inequalities, which have found versatile applications in other parts of physics and mathematics, e.g. in analysis of Navier-Stokes equations [Con88, Lie02], and studies of ionization of atoms [Lie01a, Nam12]. They constitute interesting tools on their own, and therefore warrant even further study. This part of the thesis therefore treats one type of generalization of Theorem 8, by rst passing to matrix-valued potentials and by also including a rst order derivative term in the Hamiltonian. More specically, we will study d2 d L := − ⊗ + P (x) ⊗ + Qe(x), (3.27) dx2 I dx I 14 CHAPTER 3. LIEB-THIRRING INEQUALITIES where P and Qe are given n × n-matrix functions. The case where P = 0 has been treated in [Lap00, Ben00, Exn14] and we seek to generalize their results. This gives the following problem.

Problem 1. Under which conditions on P , Qe, and the domain of L will an analogue of Theorem 8 hold?

3.3 Summary of results

This section provides one answer to the open-ended Problem 1, solving it for all self- adjoint L bounded from below. We will study three dierent settings of the problem, where the operator L acts on functions dened on all of R, on R+, and on [0, 1], respectively. The last two cases will be endowed with Robin boundary conditions at their respective endpoints. In detail, let S, S0, S1 be Hermitian n × n-matrices. We dene three dierent domains of L by

2 n D(L) : = L (R; C ), (3.28) 2 n 0 D+(L) : = {ψ ∈ L (R+; C ): ψ (0) − Sψ(0) = 0},P (0) = 0, (3.29) D (L) : = {ψ ∈ L2([0, 1]; n): [0,1] C (3.30) 0 0 ψ (0) − S0ψ(0) = 0 = ψ (1) − S1ψ(1)},P (0) = P (1) = 0, respectively. Here, the conditions that P vanishes at the end-points of the respective intervals are necessary for L to be symmetric. In fact, one can show that L is symmetric if and only if this is the case, together with

P ∗ = −P, (3.31) Qe∗ = Qe − P 0. (3.32)

Assuming this to be true, we can then rewrite L on the form

d2 d P 0(x) P (x)2 L := − ⊗ + P (x) ⊗ + − + Q(x), (3.33) dx2 I dx I 2 4 where P 0 P 2 Q = Qe − + (3.34) 2 4 is an eective potential (and, importantly, a Hermitian matrix function). This will simplify the notation in the following results, where we establish Lieb-Thirring inequalities for all settings in Equations (3.28)-(3.30). A longer discussion, together with the proofs, can be found in [Mic15], where we also give a simple criterion for L to be bounded from below. The following theorems are the main results in the rst part of the thesis. 3.3. SUMMARY OF RESULTS 15

Theorem 9. [Mic15] Let L act on D(L) in Equation (3.28). Let also P (x) be anti- Hermitian, termwise weakly dierentiable and Q(x) Hermitian. If L is bounded from below and 2 1 , then Tr(Q−) ∈ L (R) ∞ Z X 3/2 3 κ λ ≤ Tr(Q2 )dx, (3.35) i i 16 − i=1 R where κi denotes the multiplicity of the negative eigenvalue −λi. This bound is also sharp, in the sense that there are P,Q such that the resulting inequality is false if the constant is replaced by a smaller number.

Theorem 10. [Mic15] Let P (x) be anti-Hermitian, termwise weakly dierentiable with P (0) = 0 and Q(x) Hermitian. If L acts on D+(L) in Equation (3.29), is bounded from below and if 2 1 , then Tr(Q−) ∈ L (R+) ∞ Z 3 1 3/2 X 3/2 3 1 λ TrS + (2κ − n) λ + κ λ ≤ Tr(Q2 )dx + TrS3. (3.36) 4 1 2 1 1 i i 16 − 4 i=2 R+ Theorem 11. [Mic15] Let P (x) be anti-Hermitian, termwise weakly dierentiable with P (0) = P (1) = 0 and Q(x) Hermitian. If L acts on D[0,1](L) in Equa- tion (3.30), is bounded from below and if 2 1 , then Tr(Q−) ∈ L ([0, 1]) ∞ Z 1 3 X 3/2 3 1 λ Tr (S − S ) + κ λ ≤ Tr(Q2 )dx + Tr S3 − S3 . (3.37) 4 1 0 1 i i 16 − 4 0 1 i=2 0 Corollary 2. [Mic15] Let L act on D(L) in Equation (3.28) and take γ ≥ 3/2. Let P (x) be anti-Hermitian, termwise weakly dierentiable and Q(x) Hermitian. If L is γ+ 1 bounded from below and 2 1 , then Tr(Q− ) ∈ L (R) ∞ Z γ+ 1 X γ 2 (3.38) κiλi ≤ Lγ,1 Tr(Q− )dx. i=1 R These bounds are also sharp.

Corollary 3. [Mic15] Take γ ≥ 3/2 and assume that TrS3 ≤ 0. Let P (x) be anti- Hermitian, termwise weakly dierentiable with P (0) = 0 and Q(x) Hermitian. If L γ+ 1 acts on in Equation (3.29), is bounded from below and if 2 1 , D+(L) Tr(Q− ) ∈ L (R+) then ∞ 3 B(γ − 3/2, 2) γ−1/2 1 X λ TrS + (2κ − n) λγ + κ λγ (3.39) 4 B(γ − 3/2, 5/2) 1 2 1 1 i i i=2 Z ∞ γ+1/2 (3.40) ≤ Lγ,1 Tr Q− dx, 0 where B(p, q) denotes the Beta function Z 1 B(p, q) = (1 − t)q−1tp−1dt. (3.41) 0 16 CHAPTER 3. LIEB-THIRRING INEQUALITIES

Corollary 4. [Mic15] Take and assume that 3 3 . Let γ ≥ 3/2 Tr S0 − S1 ≤ 0 P (x) be anti-Hermitian, termwise weakly dierentiable with P (0) = P (1) = 0 and Q(x) Hermitian. If L acts on D[0,1](L) in Equation (3.30), is bounded from below and if γ+ 1 2 1 , then Tr(Q− ) ∈ L ([0, 1])

∞ Z 1 3 B(γ − 3/2, 2) γ−1/2 X γ+1/2 λ Tr (S − S ) + κ λγ ≤ L Tr Q dx. 4 B(γ − 3/2, 5/2) 1 0 1 i i γ,1 − i=2 0 (3.42)

3.4 Examples, remarks and comments

We now illustrate the results in the preceding section by a few examples.

Example 1. Let n = 1, P = Q = S0 = 0 and let us study the case of functions dened on the interval [0, 1]. All eigenfunctions ψ with eigenvalue −λ ≤ 0 to the operator in Equation (3.33) then satisfy √ √ 0 sinh( λ) ψ (1) = λ √ ψ(1) = S1ψ(1). (3.43) cosh( λ) √ √ sinh( λ) The function λ 7→ λ √ is a bijection from to , so there is a unique cosh( λ) R+ R+ eigenfunction for any S1 ≥ 0. The bound in Theorem 11 then becomes √ 3 3 √ sinh( λ) λTr (S0 − S1) = − λ λ √ (3.44) 4 4 cosh( λ) √ !3 1 1 √ sinh( λ) ≤ Tr S3 − S3 = − λ √ , (3.45) 4 0 1 4 cosh( λ) i.e.

√  " √ #2 3/2 sinh( λ) sinh( λ) λ √ 3 − √  ≥ 0, (3.46) cosh( λ) cosh( λ) which is clearly true.

Example 2. By letting S1 → 0 in the example above, the inequality becomes an equality.

Example 3. One might hope to use Theorem 9 to exclude the existence of negative

0 2 γ+1/2 eigenvalues. Note that γ+1/2 P P for some would Q− = Qe − + = 0 γ ≥ 3/2 2 4 − imply that our operator L in Equation (3.33) has no negative eigenvalues. However, 0 2 this is true precisely when Qe − P + P = 0. To see this, note that under our 2 4 − 3.4. EXAMPLES, REMARKS AND COMMENTS 17

assumptions on P and Qe, the matrix Q− is Hermitian. We can then diagonalize ∗ Q−(x) = U(x) D(x)U(x), where   µ1(x)− 0 0 ··· 0  0 µ2(x)− 0 ··· 0   . . .   . . .. .  (3.47) D(x) =  . . . .  ,    0 0 . . . µn−1(x)− 0  0 0 ... 0 µn(x)− has diagonal equal to the negative eigenvalues of Q(x), and U(x) is unitary. If γ+1/2 ∗ γ+1/2 is zero, then Q− = U(x) D(x) U(x)

 γ+1/2  µ1(x)− 0 0 ··· 0  0 µ (x)γ+1/2 0 ··· 0   2 −  γ+1/2  . . . .  D(x) =  . . .. .  , (3.48)    γ+1/2   0 0 . . . µn−1(x)− 0  γ+1/2 0 0 ... 0 µn(x)−

∗ must be zero. This implies D(x) = 0, so Q− = U(x) D(x)U(x) = 0, and the claim is proven.

Example 4. The main parts of the proofs of the theorems in Section 3.3 consist in making the substitution ψ = Ψψe for some appropriate matrix function Ψ. This can physically be interpreted as a generalized gauge transformation. Together with the conditions on P, Qe in Equations (3.31) and (3.32), this allows us to reformulate the spectral problem into one that is well-known. This substitution can also be of interest when studying not necessarily self-adjoint operators of the form

d2 H := − + q(x), (3.49) 1 dx2 acting on functions dened on R. Here, q is a complex-valued scalar function, and this situation was studied e.g. in [Abr01]. There, the authors showed that all eigenvalues λ 6∈ R+ to H1 satisfy 2 |λ| ≤ kqk 1 /4, (3.50) L (R) 1 2 provided that q ∈ L (R) ∩ L (R). Note that these eigenvalues will in general be complex-valued. However, by adding to this operator an appropriate term of the form d , we can ensure that the resulting operator p(x) dx d2 d H := − + p(x) + q(x) (3.51) 2 dx2 dx only has real eigenvalues. The physical interpretation of this would be that an added generalized magnetic eld lifts the non-self-adjointness. This procedure will work for certain conditions on q; in fact, we have the following result. 18 CHAPTER 3. LIEB-THIRRING INEQUALITIES

2 4 Proposition 1. Take q(x) = h(x)+ik(x). If there is a positive s ∈ L (R)∩L (R)∩ 2 C (R) s.t. Z x k(y) 2 4 Is,k(x) := s(x) dy ∈ L (R) ∩ L (R), (3.52) 0 s(y) s0(x) ∈ L2( ) ∩ L4( ), (3.53) s(x) R R s00(x) ∈ L1( ) ∩ L2( ), (3.54) s(x) R R then there is a p : R → C s.t. d2 d H = − + p(x) + q(x) (3.55) 2 dx2 dx only has real eigenvalues λ. The negative eigenvalues of H2 all satisfy

2 0 2 p p 4 |λ| ≤ q + − . (3.56) 4 2 L1(R) Proof. Take s0(x) f(x) := . (3.57) s(x) Then R x 2 4 , so take exp ( 0 f(y)dy) = s(x) ∈ L (R) ∩ L (R) Z x Z x Z y g(x) := exp f(y)dy 1 + 2 k(y) exp − f(t)dt dy (3.58) 0 0 0 = s(x) + Is,k. (3.59)

2 4 0 By assumption, g ∈ L (R)∩L (R), and dierentiating g results in g (x)−f(x)g(x) = 2k(x). If we take p := f + ig, then

fg g0 p2 p0 0 = k + − = Im q + − . (3.60) 2 2 4 2

Applying the generalized gauge transformation to H2 in this scalar case shows that it has the same eigenvalues as the operator

d2 p(x)2 p0(x) − + q(x) + − . (3.61) dx2 4 2 The details of this calculation can be found in [Mic15]. Equation (3.60) now shows that the gauge-transformed operator has real potential part. It is therefore symmetric and we have

0 g ≤ kfgk 1 + 2 kkk 1 ≤ kfk 2 kgk 2 + 2 kkk 1 , (3.62) L1(R) L (R) L (R) L (R) L (R) L (R) 0 2 2 g ≤ kfgk 2 + 2 kkk 2 ≤ kfk 4 kgk 4 + 2 kkk 2 . (3.63) L2(R) L (R) L (R) L (R) L (R) L (R) 3.4. EXAMPLES, REMARKS AND COMMENTS 19

Together with the assumptions in Equation (3.52), this guarantees that the potential 1 2 of the gauge-transformed operator is in L (R) ∩ L (R). Applying the result from [Abr01] cited above, we conclude the desired bounds on the eigenvalues.

An example of a function satisfying the last two assumptions in Equations (3.53) and (3.54) is e.g. s(x) decaying as 1 for β ≥ 1, when |x| is large. The rst |x|β assumption is satised e.g. when k(x) 1 , since then s(x) ∈ L (R)

Z Z Z x 2 2 2 k(y) 2 2 |I | dx = |s(x)| dy dx ≤ ksk 2 · kk/sk 1 , (3.64) s,k L (R) L (R) R R 0 s(y) Z Z Z x 4 4 4 k(y) 4 4 |I | dx = |s(x)| dy dx ≤ ksk 4 · kk/sk 1 . (3.65) s,k L (R) L (R) R R 0 s(y) Example 5. The following is a formal example of how the bounds in Corollary 4 relate to those in Corollary 2. It is included for the sake of physical intuition and is not intended to be mathematically rigorous. In a special case, we will see that Corollary 2 can be seen as a consequence of Corollary 4, albeit with less tight control in the resulting bounds. We now deal with trapped particles, i.e. quantum mechanical systems conned to the bounded interval I := [0, 1] of the real line, when subjected to a potential V . This can physically be modelled by forming a potential ( V (x), x ∈ I, Va(x) = (3.66) a ⊗ I, x 6∈ I, for some positive . Solving the associated Schrödinger equation shows that its so- a √ lutions have components of the form ψ(x) = Ce± a−λx outside I, for all eigenval- 2 ues λ ≤ a. Formally letting a → ∞, this means that all L (R)-solutions to the Schrödinger equation with the formal potential ( V (x), x ∈ I, V∞(x) = (3.67) ∞ ⊗ I, x 6∈ I, should be interpreted as being zero on R r I. Assuming that the Lieb-Thirring inequalities are continuous under this limiting procedure, we obtain

∞ Z γ+ 1 Z γ+ 1 X γ 2 2 (3.68) κiλi ≤ Lγ,1 Tr((V∞)− )dx = Lγ,1 Tr(V− )dx. i=1 R I Another way of modelling this physical scenario is contained in our setting of the interval. If we put S0 = S1 to be invertible, we fulll the conditions of Corollary 4. Here, the associated solutions to the Schrödinger equation have components which satisfy −1 0 and −1 0 . If we put and ψ(0) = S0 ψ (0) u(1) = S1 u (1) S0 = r ⊗ I = S1 formally let r → ∞, we obtain the boundary conditions ψ(0) = 0 = ψ(1), i.e. the 20 CHAPTER 3. LIEB-THIRRING INEQUALITIES

Quotient for LT−bounds Quotient for LT−bounds

35 3.5

30 3 25

20 2.5 Q 15 Q 2

10 1.5 5

0 1 20 20

16.3 2220.66 16.3 2220.66 12.6 1825.88 12.6 1973.92 8.9 1431.09 8.9 1727.18 1036.31 1480.44 5.2 5.2 641.524 1233.7 1.5 246.74 1.5 986.96 γ M γ M

(a) Q as a function of M, γ. (b) Q for an asymptotic range of parameters. Figure 3.1: Numerical tightness-check for Lieb-Thirring inequalities. same as in the case above. Again, assuming that our Lieb-Thirring inequalities are continuous in this limit, we obtain

∞ Z 1 X γ γ+1/2 (3.69) κiλi ≤ Lγ,1 Tr V− dx. i=2 0 This bound is of the same form as in Equation (3.68), but with the ground state excluded. As a simple numerical example of this formal setting, we consider the case of a square potential well, i.e. we put P = 0 and V (x) = −M for x ∈ [0, 1]. Here, we consider the formal boundary values ψ(0) = 0 = ψ(1). Solving the Schrödinger equation results in a discrete spectrum of negative eigenvalues of the form λ = n j ko n M − n2π2, for n ∈ 1, 2,..., pM/π2 . In Figure 3.1(a), we plot the quotient

!−1 Z 1 ∞ γ+1/2 X γ (3.70) Q := Lγ,1 Tr V− dx · κiλi , 0 i=2 as a function of the parameters M ≥ 0, γ ≥ 3/2. The quotient is always greater than one and Figure 3.1(b) shows the parameter interval in which it asymptotically tends to one. 4 Operators from Conformal Field Theory

4.1 Introduction and problem statement

This chapter investigates spectral questions arising in conformal eld theory, which describes quantum eld theories invariant under conformal transformations [Blu09]. We will study a family of operators from two-dimensional conformal eld theory, which model phenomena in topological string theory and in quantum Liouville theory. The physical backgrounds are technically involved, and will not be elaborated upon. Interested readers are instead referred to e.g. [Fad86, Gra14], and the remainder of the thesis will treat only spectral-theoretical notions. We will be interested in qualitative and quantitative spectral properties, together with concrete examples. The rst of these items includes e.g. criteria for discreteness of spectra, and the second consists in bounds on the number and the moments of eigenvalues of the operators. Because of the immediate physical applications, concrete examples also give valuable information and will be discussed below. The remainder of this chapter is organized as follows. We rst dene the family of operators under consideration and introduce some auxiliary notation. Sec- tion 4.2 then considers a detailed example of a generalization of an operator treated in [Fad15]. We lastly turn to a more general class of operators in Section 4.3, and study the asymptotic behaviour of the distribution of eigenvalues as well as establish a number of spectral bounds.

Denition of operators

0 2 Let ω be an imaginary number. We study functions in L (R) and dene an operator U with inverse U −1 by

(Uψ)(x) := ψ(x + 2ω0), (4.1) U −1ψ (x) := ψ(x − 2ω0). (4.2)

21 22 CHAPTER 4. OPERATORS FROM CONFORMAL FIELD THEORY

With these denitions, the free operator

−1 H0 := U + U (4.3) can be seen as a kind of discretized Laplacian, since H − 2 0 ψ (x) → ψ00(x), (4.4) (2ω0)2 0 as ω → 0, for ψ analytic at x. Including some potential function V : R → R, we will therefore consider the discretized Schrödinger operator dened by

H := H0 + V. (4.5) In the applications to conformal eld theory treated here, it is common to follow [Zam96] and adopt the parametrization 0 ib , for some , and to ω = 2 b ∈ R+ := [0, ∞) introduce the auxiliary constant i . We will use this in the following, together ω = 2b with the resulting normalization condition 0 1 . We also dene 00 0. ωω = − 4 ω = ω + ω Domains and notation

2 We dene the domain of U to be those ψ ∈ L (R) which can be continued analytically 0 2 into the set {x + iy : x, y ∈ R, 0 < y < 2 |ω |}, with ψ(x + iy) ∈ L (R) for 0 ≤ y < 2 |ω0|. We also require that the limit ψ x + 2ω0 − i0 := lim ψ x + 2ω0 − iε (4.6) ε&0 2 −1 2 exists in L (R). Likewise, the domain of U is those ψ ∈ L (R) with analytic 0 continuation into the strip {x + iy : x, y ∈ R, −2 |ω | < y < 0}, wherein ψ(x + iy) ∈ 2 L (R). Again, we require that ψ x − 2ω0 + i0 := lim ψ x − 2ω0 + iε (4.7) ε&0 2 exists in L (R). The domain of multiplication by V is naturally dened to be 2 2 D(V ) := ψ ∈ L (R): V (x)ψ(x) ∈ L (R) . (4.8) It can be veried by computation that U and V are self-adjoint on their respective domains. This can also be seen by using the functional calculus form of the spectral theorem, since d U = exp 2ω0i −i , (4.9) dx shows that is a real function of the self-adjoint operator d . U −i dx We will in the following use the Fourier-transform to obtain spectral information, and will adopt the convention of the Fourier-transform given by Z (Fψ)(p) := e−2πipxψ(x)dx, (4.10) R Z F −1ψb (x) := e2πipxψb(p)dp. (4.11) R 4.2. EXAMPLE: SPECTRAL PROPERTIES OF Hµ 23

Remark 1. For later use, we rewrite the domain D(U) ∩ D(U −1). By the Paley- 2 Wiener theorem [Ree75], this set consists precisely of those functions ψ ∈ L (R) with πip 2 . Any −1 then satises cosh ω ψb(p) ∈ L (R) ψ ∈ D(U) ∩ D(U )

Z πip 2 Z 2 2 2k (4.12) cosh ψb(p) dp ≥ Ck |p| ψb(p) dp, R ω R for all positive integers k, and some Ck > 0 which are independent of ψ. It then −1 k follows that D(U)∩D(U ) ⊆ H (R), for all k. By Corollary 1, we then in particular ∞ p have ψ ∈ C (R), so ψ ∈ L (R), for 2 ≤ p ≤ ∞.

4.2 Example: spectral properties of Hµ Retaining the notation from Section 4.1, we will consider a potential given by

πix V (x) = e ω . (4.13)

The associated discretized Schrödinger operator will be denoted by

0 0 0 πix H ψ(x) := ψ(x + 2ω ) + ψ(x − 2ω ) + e ω ψ(x), (4.14) on the domain specied in Section 4.1. Originally dened in [Fad86], this operator has applications in representations of quantum groups [Pon01], and in quantum Teichmüller theory [Kas01a, Kas01b], where related dierence equations can be re- duced to this form. It is therefore of interest to study the spectral properties of this operator. In [Kas01a, Kas01b], it was shown to have absolutely continuous spectrum [2, ∞), and in [Fad15], explicit formulas were given for its resolvent, Jost solutions, and eigenfunction expansion. In this section, we consider a generalization of the operator in Equation (4.14), by including a rst-derivative term representing a magnetic eld. This is in analogue with the procedure in Chapter 3, and leads to an interesting mathematical structure. More specically, we dene a new operator Hµ by

Hµψ(x) := ψ(x + 2ω0) + ψ(x − 2ω0) (4.15) 0 0 0 πix + 2iµω ψ(x + 2ω ) − ψ(x − 2ω ) + e ω ψ(x). Here, the term in brackets is a discretization of a rst-derivative term, which physically models a constant magnetic eld with strength µ. We will also refer to the operator

µ 0 0 0 0 0 (4.16) H0 ψ(x) := ψ(x + 2ω ) + ψ(x − 2ω ) + 2iµω ψ(x + 2ω ) − ψ(x − 2ω ) as the free operator. Our focus will be to derive detailed spectral properties of Hµ in Equation (4.15) as a function of the coupling constant µ, by following the techniques in [Fad15]. We will show that the spectra for the free operators are 24 CHAPTER 4. OPERATORS FROM CONFORMAL FIELD THEORY qualitatively distinct for the three cases 2 |µω0| < 1 (weak magnetic eld), 2 |µω0| = 1 (critical magnetic eld), and 2 |µω0| > 1 (strong magnetic eld), respectively. This corresponds to an interplay of the rst derivative term and the second derivative term in Equation (4.16), with µ assuming the spectrum of the dominating term. We will H0 nd these spectra, and compute explicit formulas for the generalized eigenfunctions for the non-free operator, in all these three cases. Where successful, formulas for Jost solutions and resolvents will be calculated, and we will also derive appropriate eigenfunction expansion theorems.

Origin of operators

Our operators are constituted by instances of the Weyl-operators U(u),V (v), which 2 act on L (R) for u, v ∈ C. These are dened by [U(u)ψ](x) = ψ(x − u), (4.17) [V (v)ψ](x) = e−ivxψ(x), (4.18) and satisfy the commutation relation

U(u)V (v) = eiuvV (v)U(u). (4.19) By the Stone-von Neumann theorem [Ree72, Tak08], this commutation relation char- acterizes the operators U and V up to unitary equivalence and has been historically important in formulating the foundations of quantum mechanics [Ros04]. In our case, we dene two operators as

[Uψ](x) = ψ(x + 2ω0), (4.20) πix [V ψ](x) = e ω ψ(x), (4.21) and note that we can then write

Hµ = U + U −1 + 2µiω0[U − U −1] + V. (4.22)

Remark 2. Note that for analytic ψ, we have Hµ − V − 2 lim ψ(x) = −ψ00(x) + iµψ0(x), (4.23) ω0→0 (2ω0)2 so Hµ is indeed a discretized version of the Schrödinger operator in the presence of a constant magnetic eld. We note for use with the algebra arising below that our specic choice of V gives

FUF −1 = V −1, FV F −1 = U −1, (4.24) and remark that our operator can also be written as

Hµ = (1 − bµ)U + (1 + bµ)U −1 + V. (4.25) 4.2. EXAMPLE: SPECTRAL PROPERTIES OF Hµ 25

The modular quantum dilogarithm A special function, termed the modular quantum dilogarithm, was shown to be linked to the spectral properties of H0 in [Rui97, Kas01a, Kas01b, Der14, Fad15]. This function will be important also in this note, and is dened by

Z itz 1 e dt (4.26) γ(z) = exp − 0 . 4 R sin(ωt) sin(ω t) t 0 Here, z ∈ C has Im(z) ≤ |ω + ω | and the contour of integration avoids the singu- larity at t = 0 by an indentation into the upper complex plane. γ(z) has a number of important algebraic properties that will be used in the following and are listed below; they are by now practically standard to use in connection with the spectral properties of operators like H0. Proofs and additional properties can be found in e.g. [Kas01a, Fad01, Vol05], and the formulations below are taken from [Fad15].

Lemma 1. [Fad15] The quantum dilogarithm has the following properties.

1. γ(z) can be continued meromorphically to all of C. It has poles at −(2m + 0 1)ω − (2n + 1)ω , for m, n ∈ N. If we take π 1 β = τ + , (4.27) 12 τ then we also have i( π −β) e 4 γ(z − ω00) = + O(1), (4.28) 2πz as z → 0.

2. γ(z) satises

0 −πi z 0 γ(z + ω ) = 1 + e ω γ(z − ω ), (4.29)

−πi z γ(z + ω) = 1 + e ω0 γ(z − ω). (4.30)

The main use of the quantum dilogarithm in this context comes from the fact that it can be used to describe eigenfunctions to H0 in the momentum-basis. By applying the Fourier-transform to the eigenvalue equation of Equation (4.14), we obtain πip FH0F −1ψb (p) = ψb(p + 2ω0) + 2 cosh ψb(p) = λψb(p). (4.31) ω

This was shown in [Kas01a, Kas01b] to have a generalized solution for λ ∈ [2, ∞), by using the parametrization πik . Real then correspond to 0 λ = 2 cosh ω k λ ∈ σ(H ) and k with 0 < Im(k) ≤ |ω| correspond to λ ∈ ρ(H0). Equation (4.31) is explicitly solved by −iπ(p−ω00)2 00 00 ϕ0(p, k) = c(k)e γ(p − k − ω )γ(p + k − ω ). (4.32) 26 CHAPTER 4. OPERATORS FROM CONFORMAL FIELD THEORY

Here, c(k) = e−iβ−iπk2 is a phase normalization constant. We will use and modify this formula in what follows below. We next investigate the spectrum of Hµ and divide into three cases. The analysis of each section is similar and therefore structured in the same way.

Weak magnetic eld: |bµ| < 1 Spectrum of Hµ Proposition 2. Hµ is unitarily equivalent to p1 − b2µ2H0.

Proof. Let a, b be real numbers and dene

0 µ Ta,b : D(H ) → D(H ) (4.33) ψ(x) 7→ eiaxψ(x + b). (4.34)

Integrating by parts, we see that Ta,b is unitary. If we now choose

1 1 + bµ a = ln ∈ , (4.35) ∗ 4iω0 1 − bµ R ω b = − ln 1 − b2µ2 ∈ , (4.36) ∗ 2πi R then

HµT ψ(x) = (1 − bµ)UT ψ(x) + (1 + bµ)U −1T ψ(x) a∗,b∗ a∗,b∗ a∗,b∗ (4.37)

+ VTa∗,b∗ ψ(x) ia x 2iω0a 0 πix ia x = e ∗ (1 − bµ)e ∗ ψ(x + b + 2ω ) + e ω e ∗ ψ(x + b) (4.38) 0 + eia∗x(1 + bµ)e−2iω a∗ ψ(x + b − 2ω0) p 2 2 0 (4.39) = 1 − b µ Ta∗,b∗ H ψ(x).

Note that this in particular implies that p p σ(Hµ) = 1 − b2µ2σ(H0) = [2 1 − b2µ2, ∞), (4.40) with multiplicity two. Next, in order to obtain explicit formulas for the remaining spectral properties of interest, we apply this to the results in [Fad15]. We emphasize that all calculations below reduce to those of [Fad15] if we put µ = 0.

Resolvent and eigenfunctions of the free operator Applying the Fourier-transform to the eigenvalue equation corresponding to Equa- tion (4.16), we obtain µ −1 (4.41) FH0 F ψb (p) = f(p)ψb(p), 4.2. EXAMPLE: SPECTRAL PROPERTIES OF Hµ 27

h i for πip πip . We can also rewrite f(p) := 2 cosh ω + bµ sinh ω

πip πip f(p) = A cosh + θ = A cosh e , (4.42) ω ω where we introduced the auxiliary parameters p A = 2 1 − b2µ2, (4.43) 1 cosh (θ) = , (4.44) p1 − b2µ2 bµ sinh (θ) = , (4.45) p1 − b2µ2 ωθ p = p + . (4.46) e πi The resolvent at λ ∈ C is therefore a bounded operator in the momentum representation precisely when λ 6∈ Ran(f) = [2p1 − b2µ2, ∞). We can then form

Z e2πipx R0(x; λ) = dp, (4.47) R f(p) − λ which gives the resolvent to be Z (R(λ)ψ)(x) = R0(x − y; λ)ψ(y)dy, (4.48) R after applying the inverse Fourier-transform. We now parametrize the spectrum by writing πiek , for ωθ . Here, corresponds λ = f(k) = A cosh ω ek = k + πi k ∈ R to µ , which shows that the spectrum has multiplicity two. A in the λ ∈ σ(H0 ) λ resolvent-set corresponds to a k with 0 < Im(k) ≤ |ω|. With this parametrization, we note that the integrand in Equation (4.47) has singularities at p = k + 2nω and 2ωθ , for integers. We can then compute the integral in p = −k − πi + 2mω n, m Equation (4.47) using the residue theorem, and obtain

" −2πi(k+ 2ωθ )x 2πikx # 2ω e πi e R0(x; λ) = + (4.49) πiek 1 − e−4πiωx 1 − e4πiωx A sinh ω " # 2ωeia∗x e−2πiekx e2πiekx = + . (4.50) πiek 1 − e−4πiωx 1 − e4πiωx A sinh ω

If we put µ = 0, this result is consistent with [Fad15]. With our parametrization of λ, we can also nd the generalized eigenfunctions directly. Using Equa- 2πikx −2πi(k+ 2ωθ )x tion (4.16), we see that f+(x, k) := e , f−(x, k) := e πi both satisfy µ . These are therefore analogous to the Jost solutions for H0 f±(x, k) = λf±(x, k) Schrödinger operators. 28 CHAPTER 4. OPERATORS FROM CONFORMAL FIELD THEORY

Eigenfunctions of Hµ We retain the notation of Section 4.2 and study the eigenvalue problem in the momentum-basis. Fourier-transforming Equation (4.15), we obtain the associated eigenvalue equation

πip ψb(p + 2ω0) + A cosh e ψb(p) = λψb(p). (4.51) ω Parametrizing πiek as in the last section, we compare this equation to λ = A cosh ω Equation (4.31). An explicit solution to Equation (4.51) can then be written down as

00 2 2ωθb∗ 2πipb∗ −iπ(p−ω ) 00 00 (4.52) ϕbµ(p, k) = e cµ(k)e e e γ(pe+ ek − ω )γ(pe− ek − ω ).

2 0 In this equation, −iβ−iπek and ω 2 2, so that 4πiω b∗ cµ(k) = e b∗ = − 2πi ln 1 − b µ e = A/2. That this is indeed a solution to Equation (4.51) follows from a straightforward calculation using property two in Lemma 1. Since the function πiek k 7→ A cosh ω for 0 ≤ Im(k) ≤ |ω| is a 2 − 1 mapping of the complex plane, it follows again that the spectrum above has multiplicity two. Returning to the coordinate-representation, we dene Z 2πipx (4.53) ϕµ(x, k) = e ϕbµ(p, k)dp R Z 00 2 2ωθb∗ 2πipx 2πipb∗ −iπ(p−ω ) 00 00 = e e cµ(k)e e e γ(pe+ ek − ω )γ(pe− ek − ω )dp R (4.54)

Z 00 2 2ωθb∗ 2πip(x+b∗) −iπ(p−ω ) 00 00 = e e e c0(ek)e e γ(pe+ ek − ω )γ(pe− ek − ω )dp R (4.55) Z 2ωθb∗ −2ωθ(x+b∗) 2πip(x+b∗) (4.56) = e e e e ϕb0(p,e ek)dpe R 2ωθb∗ −2ωθ(x+b∗) ia∗x = e e ϕ0 x + b∗, ek = e ϕ0 x + b∗, ek . (4.57)

The last equality used the fact that ia∗ , as can be seen from the denitions θ = − 2ω µ above. ϕµ then satises H ϕµ(x, k) = λϕµ(x, k).

Jost solutions The Jost solutions (µ) of µ can be obtained from the corresponding Jost f± (x, k) H solutions (0) of 0, using Proposition 2. f± (x, k) H Proposition 3. The functions (µ) ia∗x∓2πiekb∗ (0) (4.58) f± := e f± x + b∗, ±ek 4.2. EXAMPLE: SPECTRAL PROPERTIES OF Hµ 29 satisfy ! (µ) πiek (µ) Hµf = A cosh f , (4.59) ± ω ± and

(µ) 2πikx (4.60) f+ = e + o(1), (µ) 2ωθ −2πi(k+ iπ )x (4.61) f− = e + o(1), as x → −∞. Proof. We only perform the straightforward calculations for (µ). The case (µ) is f+ f− similar. We rst calculate

µ (µ) A 0 −1 (µ) H f = Ta ,b H T f (4.62) + ∗ ∗ 2 a∗,b∗ + ! A πik ia∗x−2πiekb∗ e (0) (µ) = e 2 cosh f x + b∗, ek = λf . (4.63) 2 ω + +

Next, it was shown in [Fad15] that (0) 2πiekx as , so f+ (x, ek) = e + o(1) x → −∞

(µ) 2πiekx+ia∗x 2πikx as (4.64) f+ (x, k) = e + o(1) = e + o(1), x → −∞. This concludes the proof.

(µ) therefore behave asymptotically like the generalized eigenvectors of µ as f± H0 x → −∞, and so play the role of Jost solutions.

Resolvent of Hµ By Proposition 2, we can express the resolvent of Hµ as −1 µ −1 A 0 −1 Rµ(λ) := (H − λ) = Ta ,b H − λ T (4.65) ∗ ∗ 2 a∗,b∗ −1 2 0 2 −1 = Ta ,b H − λ T . (4.66) A ∗ ∗ A a∗,b∗ To calculate this, we will use the results from [Fad15] showing that Z 0 −1 H − λ ψ (x) = R0(x − y; k)ψ(y)dy, (4.67) R where πik and λ = 2 cosh ω

" (0) (0) # ω f− (x, k)ϕ0(y, k) f− (y, k)ϕ0(x, k) R0(x; k) = + . (4.68) πik πix − πix sinh ω 1 − e ω0 1 − e ω0 30 CHAPTER 4. OPERATORS FROM CONFORMAL FIELD THEORY

Using this in Equation (4.66), we obtain

2 Z (R (λ)ψ)(x) = eia∗x R (x + b − y; k)T −1 ψ(y)dy (4.69) µ 0 ∗ e a∗,b∗ A R Z 2 ia∗(x−y+b∗) = e R0(x + b∗ − y; ek)ψ(y − b∗)dy (4.70) A R Z 2 ia∗(x−y) = e R0(x − y; ek)ψ(y)dy, (4.71) R A i.e. Rµ(λ) is the integral operator with kernel

ia (x−y) " (0) (0) # 2ωe ∗ f (x, ek)ϕ0(y, ek) f (y, ek)ϕ0(x, ek) − + − . (4.72) πix − πix πiek ω0 ω0 A sinh ω 1 − e 1 − e

2 Remark 3. This is a bounded operator on L (R), since it is equal to the bounded operator R0(x; λ), up to a phase term and a shift of the k-argument.

Remark 4. Note that |a∗| → ∞ as |bµ| → 1. The solutions to Equation (4.51) are therefore not continuous across the phase transition at |bµ| = 1. We will see below that this is accompanied by an associated change in the spectrum of µ. H0

Eigenfunction expansion

In the case µ = 0, it was shown in [Fad15] that

2 2 U : L (R) → L ([0, ∞), ρ(k)dk) (4.73) Z ψ 7→ ψ(x)ϕ0(x, k)dx, (4.74) R where πik πik is an isometric bijection. Moreover, the operator ρ(k) = 4 sinh ω sinh ω0 0 −1 acts by multiplication by πik . In our case, Proposition 2 gives UH U 2 cosh ω Hµ = A T H0T −1 , i.e. 2 a∗,b∗ a∗,b∗ −1 A UT −1 Hµ UT −1 = UH0U −1 (4.75) a∗,b∗ a∗,b∗ 2 acts by multiplication with πik . Since is unitary, µ is unitarily A cosh ω Ta∗,b∗ H equivalent to multiplication with πik and so in particular has spectrum A cosh ω [A, ∞). The eigenfunction expansion is therefore given by the operator Z Z UT −1 ψ (k) = T −1 ψ(x) ϕ (x, k)dx = ψ(x)ϕ (x, k)dx. (4.76) a∗,b∗ a∗,b∗ 0 µ R R

Critical magnetic eld: |bµ| = 1 We consider the case bµ = −1. The case bµ = 1 can then be obtained by the substitution b 7→ −b, and is therefore omitted. 4.2. EXAMPLE: SPECTRAL PROPERTIES OF Hµ 31

Resolvent and eigenfunctions of the free operator

πik We parametrize λ = 2e ω and apply the Fourier-transform to Equation (4.16). This gives the eigenequation πip πik 2e ω ψb = 2e ω ψ.b (4.77)

µ πip µ The spectrum of then consists of all in the range of ω , i.e. , H0 λ 2e σ(H0 ) = R+ and it has multiplicity one. A λ in the resolvent-set corresponds to a k with 0 < Im(k) ≤ |ω|. Applying the inverse Fourier-transform, we obtain the resolvent Z (R(λ)ψ)(x) = R0(x − y; λ)ψ(y)dy, (4.78) R where Z 2πipx 1 e (4.79) R0(x; λ) = πip πik dp. 2 R e ω − e ω This integral can be evaluated using the residue theorem, with the result

2πikx ω e (4.80) R0(x; λ) = πik 4ωπix . e ω 1 − e

Note lastly that the generalized eigenfunction corresponding to a k ∈ R is simply f(x) = e2πikx.

Eigenfunctions of Hµ

πik If we parametrize λ = e ω , the eigenvalue equation becomes

0 πix πik 2ψ(x + 2ω ) + e ω ψ(x) = e ω ψ(x), (4.81) which can be solved explicitly. Take i ; using property two in Lemma 1, a c = b ln(2) straightforward calculation shows that

2πikx+cx 00 ϕµ(x, k) = e γ(−x + k − ω ) (4.82)

µ πik satises H ϕµ(x, k) = e ω ϕµ(x, k).

Strong magnetic eld: |bµ| > 1 Resolvent and eigenfunctions of the free operator In this parameter regime, we Fourier-transform Equation (4.15) to obtain the eigenvalue equation µ −1 (4.83) FH0 F ψb (p) = f(p)ψb(p). 32 CHAPTER 4. OPERATORS FROM CONFORMAL FIELD THEORY

Here, πipe , where we again introduced help variables through f(p) = A sinh ω p A = 2 b2µ2 − 1, (4.84) bµ cosh (θ) = , (4.85) pb2µ2 − 1 1 sinh (θ) = , (4.86) pb2µ2 − 1 ωθ p = p + . (4.87) e πi Precisely as in the cases above, we have the resolvent Z (R(λ)ψ)(x) = R0(x − y; λ)ψ(y)dy, (4.88) R for λ ∈ C and Z e2πipx R0(x; λ) = dp, (4.89) R f(p) − λ after applying the inverse Fourier-transform. R(λ) will be a bounded operator on 2 L (R) in the momentum representation precisely when λ 6∈ Ran(f) = R. We therefore parametrize the spectrum by writing πiek , for ωθ . λ = f(k) = A sinh ω ek = k + πi Again, corresponds to µ , so the spectrum has multiplicity one. k ∈ R λ ∈ σ(H0 ) A λ in the resolvent-set corresponds to a k with 0 < Im(k) ≤ |ω|. With this parametrization, the integrand in Equation (4.89) has singularities at p = k + 2nω, 2ωθ , for integers. We can then compute the integral in p = −k − πi + (2m + 1)ω m, n Equation (4.89) using the residue theorem, and obtain

2ω 1 h 2πikx −2πix(k+ 2ωθ )+2πiωxi R0(x; λ) = e − e πi . (4.90) πiek 1 − e4πixω A cosh ω

−2πIm(k)|x−y| Note that R0(x; λ) has a limit as x → 0 and that |R0(x; λ)| ≤ e , so this 2 does indeed dene a bounded operator on L (R) by Schur's lemma. With our parametrization of λ, we can also nd the generalized eigenfunc- 2πikx tions directly. Using Equation (4.16), we see that f+(x, k) := e , f−(x, k) := 2ωθ µ −2πi(k+ πi )x+2πiωx both satisfy . We can then rewrite Equa- e H0 f±(x, k) = λf±(x, k) tion (4.90) as

2ω f (x)f (y) − f (y)f (x) R (x − y; λ) = + − + − . (4.91) 0 4πiω(x−y) (1 − bµ) C(f−, f+)(x) 1 − e Here, C(f, g) denotes the Casorati-determinant

C(f, g) := f(x + 2ω0)g(x) − f(x)g(x + 2ω0), (4.92) which is a discretized analogue of the Wronskian for dierential equations. 4.2. EXAMPLE: SPECTRAL PROPERTIES OF Hµ 33

Remark 5. For functions f, g satisfying the eigenvalue equation associated to Equa- tion (4.16), we have

C(f, g)(x) = f(x + 2ω0)g(x) − f(x)g(x + 2ω0) (4.93) 1 + bµ 1 + bµ = −f(x − 2ω0)g(x) + f(x)g(x − 2ω0) (4.94) 1 − bµ 1 − bµ 1 + bµ = C(f, g)(x − 2ω0). (4.95) 1 − bµ We will use this to directly see that

(Hµ − λ) R(λ)ψ = ψ. (4.96)

We will only need to show that R0(x; λ) satises

(1 − bµ)R (x + 2ω0 − y − i0; λ) + (1 + bµ)R (x − 2ω0 − y + i0; λ) 0 0 (4.97) −λR0(x − y; λ) = δ(x − y), in a distributional sense. This can be seen from Equation (4.91); since f± are eigenfunctions of Hµ, both sides in Equation (4.97) are zero for x 6= y. For x = y, the left hand side has singular part

0 0 −1 f+(x + 2ω )f−(y) − f+(y)f−(x + 2ω ) 2πi(1 − bµ)C(f , f )(x) x − y − i0 − + (4.98) f (x − 2ω0)f (y) − f (y)f (x − 2ω0) − + − + − x − y + i0 1 1 1 = − (4.99) 2πi x − y − i0 x − y + i0 = δ(x − y). (4.100)

Here, the last step is precisely the so-called Sokhotski-Plemelj formula, which proves Equation (4.96).

Eigenfunctions of Hµ We again apply the Fourier-transform to the eigenvalue equation for the operator in Equation (4.15), and obtain

πip ψb(p + 2ω0) + A sinh e ψb(p) = λψb(p). (4.101) ω Using the same parametrization and notation as before, we can again use property two in Lemma 1 to nd the generalized eigenfunctions. A straightforward calculation shows that these can be written as

ipB −iπ(p−ω00)2 0 00 (4.102) ϕbµ(p, k) = cµ(k)e e e γ(pe+ ek − ω )γ(pe− ek − ω ), 34 CHAPTER 4. OPERATORS FROM CONFORMAL FIELD THEORY where 1 A , and −iβ−iπek2 , as before. Note that the rst - B = − b ln 2 cµ(k) = e γ function has argument shifted by +ω as compared to the operator for the weak magnetic eld. We also dene Z 2πipx (4.103) ϕµ(x, k) = e ϕbµ(p, k)dp, R which then satises µ πiek . H ϕµ(x, k) = A sinh ω ϕµ(x, k)

4.3 Eigenvalue estimates for general potentials

This the last section returns to the case of a general potential function for the discretized Schrödinger operator in Equation (4.5). In order to gain a better understanding of the operator H, we would like to obtain more quantitative information on the magnitude of its eigenvalues. A rst problem is to understand when H has discrete spectrum. When this is the case, we will denote its eigenvalues by ∞ . The remainder of the chapter {λn}n=1 will then study two quantities; the rst of these is the eigenvalue counting function N(Λ,H), dened by

N(Λ,H) := |{fn ∈ D(H): Hfn = λnfn, with λn ≤ Λ}| , (4.104) where |·| is the counting measure. The second quantity is the Riesz-mean, or eigenvalue-moment, which is dened as

∞ X γ |λn| , (4.105) n=1 for some γ ∈ R+. Both N(Λ,H) and the eigenvalue-moments are physical quantities, as seen e.g. in the Lieb-Thirring inequalities of Chapter 3. Our goal is to estimate these quantities in terms of the potential V , which oers general spectral information without necessitating an explicit computation of the eigenvalues.

Discreteness of spectrum For the continuous Schrödinger operator

− ∆ + V (x), (4.106)

d acting on functions dened on R , a classical and sucient condition for the corresponding spectrum to be discrete is that V (x) → +∞ when |x| → ∞, as shown in [Fri34]. A similar result was proven in [Sim09], originally for the continuous Schrödinger operator, but also generalized to a wider class of operators. This section applies this condition also to the operator

−1 H := H0 + V = U + U + V, (4.107) 4.3. EIGENVALUE ESTIMATES FOR GENERAL POTENTIALS 35 where V (x) is real-valued and essentially bounded from below. We could show that our operator is of the kind permitted in [Sim09] by redening it in terms of quadratic forms, but instead opt for adapting the proof used therein, for the sake of completeness. To do so, we will need the following.

−tH Lemma 2. The operator H0 generates a positive contraction semigroup e 0 , given by Z −tH0 (4.108) e ψ (x) = gbt(x − y)ψ(y)dy, R for πip . gt(p) = exp −2t cosh ω

Proof. Since H0 is positive and self-adjoint, the rst statement follows from the Hille-Yosida theorem [Ree75], and it only remains to show positivity and the representation as an integral operator. Note that e−tH0 ψ(x) = ψ(x, t), where the latter is given by

∂ ψ(x, t) = −H ψ(x, t), (4.109) ∂t 0 ψ(x, 0) = ψ(x). (4.110)

Applying the Fourier-transform to the x-coordinate, this is equivalent to

∂ πip ψb(p, t) = −2 cosh ψb(p, t), (4.111) ∂t ω ψb(p, 0) = ψb(p), (4.112)

−tH with the solution ψb(p, t) = gt(p)ψb(p). Since gt(p) > 0, it follows that e 0 is positive in the momentum-representation, so then also in the coordinate-representation. Taking the inverse Fourier-transform results in Z −tH0 (4.113) e ψ(x) = gbt(x − y)ψ(y)dy. R

Remark 6. We note for later use that gt(p) is in the Schwartz space, meaning that is as well, so in particular p for all . gbt gbt ∈ L (R) 1 ≤ p ≤ ∞ We can now adapt the proof from [Sim09] to prove the main result of this section.

Theorem 12 (adapted from [Sim09]). Assume that V (x) is essentially bounded from below, and that Sm := {x : V (x) ≤ m} has nite Lebesgue-measure for all m ∈ R. Then H has discrete spectrum.

Proof. One can show that H having only discrete spectrum is equivalent to e−tH being compact, for some positive t [Ree78]. Moreover, it is true that e−Ae−B being compact for some non-negative, self-adjoint operators A, B implies that e−(A+B) is 36 CHAPTER 4. OPERATORS FROM CONFORMAL FIELD THEORY compact [Sim09]. In our case, we therefore only need to prove that T := e−H0 e−V is compact, and proceed just as in [Sim09]. We decompose

T = Tm + Rm, (4.114) with , c . This gives Tm := T χSm Rm = T χSm

−H0 −m kRmk ≤ e e → 0, (4.115) as m → ∞. It follows that Tm → T in the norm sense, so we need only show that each Tm is compact to conclude that also T is. In fact, Tm is even Hilbert-Schmidt, since Z −V (y) (4.116) (Tmψ)(x) = gb1(x − y)e ψ(y)dy Sm is an integral operator, with

Z Z 2 −2V (y) 2 (4.117) g1(x − y)e dydx ≤ kg1kL2( ) · exp −2 inf V (x) |Sm| < ∞. b b R x∈ R Sm R

As this last expression is nite, Tm is compact, and so also T , which concludes the proof, by the remarks above.

From this, we can deduce the following standard result, which concludes the considerations about the discreteness of the spectrum under the conditions in The- orem 12.

2 Corollary 5. If H has discrete spectrum, then there is a complete set in L (R) consisting of orthonormal eigenfunctions to H with eigenvalues λn that satisfy λn → ∞ as n → ∞.

Proof. We rst show that eigenfunctions {fn} to H form a complete set. To see this, form X as the space spanned by the eigenfunctions and study X⊥. Now, HX⊥ ⊆ X⊥, since for any x ∈ X⊥ ∩ D(H), we have

(4.118) hHx, fniL2(R) = hx, λnfniL2(R) = 0, by the symmetry of H. Since X⊥ is a closed subset of a Hilbert space, it is also complete, and therefore a Hilbert space. This means that is a symmetric HX⊥ operator on a Hilbert space, with empty essential spectrum (inherited from H) and no eigenvalues (by denition of ⊥). is therefore empty, which is X σ(HX⊥ ) impossible unless X⊥ = {0}. 2 Next, L (R) has innite dimension, so there are innitely many fn. Their corresponding eigenvalues cannot have a nite accumulation point, since σess(H) = ∅, which shows that the set {λn} cannot be bounded. This concludes the proof. 4.3. EIGENVALUE ESTIMATES FOR GENERAL POTENTIALS 37

Asymptotic behaviour of eigenvalues

This section studies N(Λ,H), when Λ tends to innity. The main result is an asymptotic phase-space expression for N(Λ,H), which has a classical analogue in quantum mechanics and also makes an argument in [Gra14] precise. Results of this kind are inspired by Weyl [Wey11], who showed that N(Λ, −∆) for the operator −∆ d with Dirichlet boundary conditions in some bounded Ω ⊆ R , satises −d/2 −d lim Λ N(Λ, −∆) = (2π) ωd |Ω| . (4.119) Λ→∞

d Here, ωd denotes the volume of the unit ball in R . Our approach is standard and follows [Sim79, Lap97], adapted to the setting of the operator H. The main tools are Abelian-Tauberian theorems, together with the Golden-Thompson formula and the method of coherent states, which allow us to prove the following.

Theorem 13. Let α, β ≥ 0. Assume that V (x) is continuous, bounded from below, and that there exists some K ∈ R+ s.t.

Z 2 V (x)e−K|x−y| dx (4.120) R is nite, and determines a continuous function in y, for all y ∈ R. If H also has discrete spectrum, and −α −β 2 πip lim Λ (ln Λ) (p, x) ∈ R : 2 cosh + V (x) ≤ Λ = C (4.121) Λ→∞ ω holds for some C ∈ R, then lim Λ−α (ln Λ)−β N(Λ,H) = C. (4.122) Λ→∞ Remark 7. Note that the assumption that V (x) is bounded from below is implic- itly assumed in Equation (4.121), since the phase space volume would otherwise be innite for any nite Λ. It can therefore not be relaxed. The assumption in Equa- tion (4.120) is also needed in our proof; without it, we can, however, still conclude the weaker statement that

Λ−α (ln Λ)−β N(Λ,H) ≤ C(1 + o(1)), (4.123) for all Λ large enough. Remark 8. This result can be compared to the analogue for the continuous Schrödinger equation, where the term πip is replaced by the kinetic energy term 2. We 2 cosh ω p therefore interpret the hyperbolic term as analogous to kinetic energy. The physical interpretation of the theorem is then that the number of eigenstates of H asymptotically approaches the volume of the phase-space with total energy less than a prescribed level. 38 CHAPTER 4. OPERATORS FROM CONFORMAL FIELD THEORY

The proof of this fact relies on the following lemma, which is adapted and slightly generalized from [Sim79].

Lemma 3 (Abelian-Tauberian theorem; slightly generalized from [Sim79]). Take α, β ≥ 0, and C ∈ R. If µ is a non-negative measure on R+, then C lim Λ−α (ln Λ)−β µ[0, Λ) = (4.124) Λ→∞ Γ(α + 1) is equivalent to −β Z lim tα ln t−1 e−txdµ(x) = C, (4.125) t→0 R+ where we denote the Gamma-function by Z Γ(t) = xt−1e−xdx. (4.126) R+ Proof. Assume rst that Equation (4.124) holds. Integrating by parts, we obtain

−β Z −β Z tα ln t−1 e−txdµ(x) = tα+1 ln t−1 e−txµ[0, x)dx (4.127) R+ R+ −β Z = tα+1 ln t−1 e−txxα (ln x)β x−α (ln x)−β µ[0, x) dx. (4.128) R+ If we now set y = tx, then the last term equals

Z ln y β yαe−y t R(y, t)dy, (4.129) ln t−1 R+

−α −β ln y β for y y y . By hypothesis, C , and t R(y, t) = t ln t µ[0, t ) R(y, t) → Γ(α+1) ln t−1 tends to 1 pointwise in y, as t → 0. This shows Equation (4.125). Conversely, assume that Equation (4.125) holds. Letting t = Λ−1, we have

−α −β α −1−β −1 lim Λ (ln Λ) µ[0, Λ) = lim t ln t µ[0, t ) = lim µt[0, 1), (4.130) Λ→∞ t→0 t→0

α −1−β where we dene µt(M) := t ln t µ(tM), for M ⊆ R a Borel-set. Setting dν(x) = xα−1dx, we see that Equation (4.124) is equivalent to

C lim µt[0, 1) = ν[0, 1). (4.131) t→0 Γ(α)

This will follow after showing that C weakly, i.e. that µt → Γ(α) ν

Z C Z lim f(x)dµt(x) = f(x)dν(x), (4.132) t→0 Γ(α) R+ R+ 4.3. EIGENVALUE ESTIMATES FOR GENERAL POTENTIALS 39 for all ∞ . Since −nx ∞ is dense in ∞ , it suces to show f ∈ C0 (R+) {e }n=1 C0 (R+) Equation (4.132) for f(x) = e−nx. Note that the assumption in Equation (4.125) is precisely that Z Z −x C −x lim e dµt(x) = e dν(x). (4.133) t→0 Γ(α) R+ R+ We also have Z Z −nx −ntx α −1−β e dµt(x) = e t ln t dµ(x) (4.134) R+ R+ Z n −β 1 0 ln 0 −β = e−t xt0α t ln t0−1 dµ(x), (4.135) nα ln t0−1 R+ ln n −β where 0 and t0 , as 0 . Since also t = nt ln t0−1 → 1 t → 0 Z 1 Z e−nxdν = e−xdν, (4.136) nα R+ R+ we obtain Z Z −nx C −nx lim e dµt(x) = e dν(x), (4.137) t→0 Γ(α) R+ R+ for all n ∈ N. This concludes the proof. Proof of Theorem 13. Let ∞ denote the eigenvalues of . Note that this set {λn}n=0 H is bounded from below, since H0 is positive and V (x) is bounded from below. Since Equation (4.121) is invariant with respect to equal translations of V and Λ, we can without loss of generality assume that λn ≥ 0, for all n. Dene 2 πip A(Λ) := (p, x) ∈ R : 2 cosh + V (x) ≤ Λ , (4.138) ω Z πip I(t) := exp −t 2 cosh + V (x) dxdp, (4.139) R2 ω as a shorthand notation. If we take P∞ , Lemma 3 implies that µ1 = n=0 δλn −β lim tα ln t−1 Tr e−tH = Γ(α + 1) lim Λ−α (ln Λ)−β N(Λ,H), (4.140) t→0 Λ→∞ provided either limit exists. Dening a second measure µ2 by 2 πip µ2(M) := (p, x) ∈ R : 2 cosh + V (x) ∈ M , (4.141) ω for M ⊆ R a Borel-set, we can use Lemma 3 again to conclude −β lim tα ln t−1 I(t) = Γ(α + 1) lim Λ−α (ln Λ)−β A(Λ), (4.142) t→0 Λ→∞ since the second term exists by assumption. It will therefore be enough to show that

−β −β lim tα ln t−1 Tr e−tH = lim tα ln t−1 I(t), (4.143) t→0 t→0 which we will now do. 40 CHAPTER 4. OPERATORS FROM CONFORMAL FIELD THEORY

Claim 1. Tr e−tH ≤ I(t).

Proof of Claim 1. The Golden-Thompson inequality [Gol65, Tho65] gives that

−tH − t V −tH − t V Tr e ≤ Tr e 2 e 0 e 2 , (4.144) and we will nd an expression for the right-hand side. Using Lemma 2, we can write

t t Z t t − 2 V −tH0 − 2 V − 2 V (x) − 2 V (y) (4.145) e e e ψ(x) = e gbt(x − y)e ψ(y)dy R Z = Kt(x, y)ψ(y)dy, (4.146) R

t t where − 2 V (x) − 2 V (y). Now, by [Bri88], we can compute Kt(x, y) := e gbt(x − y)e

t t Z − V −tH0 − V Tr e 2 e e 2 = Kt(x, x)dx, (4.147) R

t t provided that 1 2 , and that − 2 V −tH0 − 2 V is positive as well as Kt(x, y) ∈ Lloc(R ) e e e Hilbert-Schmidt. This rst statement follows from being in the Schwartz space gbt and that V was assumed continuous; the second statement is true, since

t t − 2 V −tH0 − 2 V −tH0 (4.148) he e e ψ, ψiL2(R) = he ϕ, ϕiL2(R),

− t V (x) −tH for ϕ(x) = e 2 ψ(x). Since e 0 was seen to be positive in Lemma 2, the − t V −tH − t V expression in Equation (4.148) is positive, so e 2 e 0 e 2 is a positive operator. −tV (x) 1 For the third statement, note that it is true if e is in L (R); if this is not the case, then I(t) = ∞, so the bound in Claim 1 is void. We can therefore return to Equation (4.147), and compute

t t Z Z − 2 V −tH0 − 2 V −tV (x) (4.149) Tr e e e = Kt(x, x)dx = gbt(0)e dx = I(t). R R Inserting this into Equation (4.144) concludes the proof of the claim.

α −1−β −tH α −1−β Claim 2. limt→0 t ln t Tr e ≥ limt→0 t ln t I(t).

Proof of Claim 2. We will use the method of coherent states, and therefore dene

2πixk Fk,y(x) = e G(x − y), (4.150)

q K −K|x|2/2 for G(x) = e . Here, the constant is chosen such that kGk 2 = π L (R) ∞ kF (·)k 2 = 1. Let {ψn} be the complete, orthogonal set of eigenfunc- k,y L (R) n=0 tions with eigenvalues ∞ guaranteed by Corollary 5, and denote by the {λn}n=0 Eν 4.3. EIGENVALUE ESTIMATES FOR GENERAL POTENTIALS 41 associated spectral measure. We have Z −tH X −tλn 2 Tr e = e |ψn(x)| dx (4.151) n R Z X −tλn = e Fk,y(p)ψn(p)F k,y(q)ψn(q) dpdqdkdy (4.152) 4 n R Z Z ∞ −tν = e hdEνFk,y(·),Fk,y(·)i dkdy. (4.153) R2 0 Note that we have R ∞ , so we can 0 hdEνFk,y(·),Fk,y(·)i = hFk,y(·),Fk,y(·)iL2(R) = 1 apply Jensen's inequality to the last term above. We obtain

Z Z ∞ −tH Tr e ≥ exp −t νhdEνFk,y(·),Fk,y(·)i dkdy. (4.154) R2 0 Here Z ∞ (4.155) νhdEνFk,y(·),Fk,y(·)i = hFk,y(·),HFk,y(·)iL2(R) 0 πik Z 2 0 = exp − eKτ /2e−2Kω (x−y)G(x − y)2dx ω R πik Z 2 0 + exp eKτ /2e2Kω (x−y)G(x − y)2dx (4.156) ω R Z + V (x)G(x − y)2dx R πik 2 Z ≤ 2 cosh eKτ /2 + V (x)G(x − y)2dx. (4.157) ω R Let now , which is nite by assumption. This implies that −a = miny∈RV (y) eKτ 2/2 (V (y) + a) ≥ 0. We add this to the exponent in the right side of Equa- tion (4.154) and obtain Z h πik 2 2 Tr e−tH ≥ exp − t 2 cosh eKτ /2 + eKτ /2 (V (y) + a) (4.158) R2 ω Z i + V (x)G(x − y)2dx dkdy (4.159) R Z h πik 2 2 ≥ exp − t 2 cosh eKτ /2 + eKτ /2 (V (y) + a) (4.160) Ω ω Z i + V (x)G(x − y)2dx dkdy, (4.161) R 2 for Ω ⊆ R an arbitrary compact set. By the continuity assumption, the function Z y 7→ V (x)G(x − y)2dx (4.162) R 42 CHAPTER 4. OPERATORS FROM CONFORMAL FIELD THEORY assumes a maximum magnitude value on Ω projected onto the y-coordinate, so we obtain Z 2 πik lim Tr e−tH ≥ lim exp −teKτ /2 2 cosh + V (y) dkdy (4.163) t→0 t→0 Ω ω Z πik = lim exp −t 2 cosh + V (y) dkdy. (4.164) t→0 Ω ω

Inserting the term tα ln t−1−β (≥ 0, for t small enough) on each side of the in- 2 equality, the right hand side converges as Ω → R . We can conclude that −β −β lim tα ln t−1 Tre−tH ≥ lim tα ln t−1 I(t), (4.165) t→0 t→0 which nishes the proof of the claim.

We have therefore shown that

−β −β lim tα ln t−1 Tr e−tH = lim tα ln t−1 I(t). (4.166) t→0 t→0 By the remarks at the beginning of the proof, the theorem is now established.

Example 6. The potential πix occurs in a model of conformal eld V (x) = 2 cosh ω theory discussed in e.g. [Gra14]. We can compute the asymptotics of N(Λ,H). Denoting

πip πix E(Λ) = (p, x) ∈ 2 : 2 cosh + 2 cosh ≤ Λ , (4.167) R ω ω we note that the maximal are ω Λ . This means that x, p ∈ E(Λ) x, p = πi acosh 2 ω Λ 2. Moreover, by symmetry, the set |E(Λ)| ≤ 4 πi acosh 2 ω Λ E0(Λ) = (p, x) ∈ 2 : |x| , |p| ≤ acosh (4.168) R πi 4 is contained in E(Λ), so

2 2 ω Λ 0 ω Λ 4 acosh = E (Λ) ≤ |E(Λ)| ≤ 4 acosh . (4.169) πi 4 πi 2 For Λ large, this reduces to

ω Λ2 ω 2 4 ln ≤ |E(Λ)| ≤ 4 ln (Λ) , (4.170) πi 2 πi and Theorem 13 results in the asymptotics

1 lim ln (Λ)−2 N(Λ,H) = . (4.171) Λ→∞ b2π2 4.3. EIGENVALUE ESTIMATES FOR GENERAL POTENTIALS 43

Spectral bounds The bound in the last section concerns only the asymptotic behaviour for large eigenvalues, and it is also of interest to obtain general bounds, valid regardless of the size of the eigenvalue. This section derives various spectral bounds, which are not necessarily sharp, but still of interest. This section does not assume that V → ∞ as |x| → ∞, so in particular does not assume that we are studying a discrete spectrum.

Non-self adjoint operators In this section, we prove a general bound for eigenvalues of the discretized Schrödinger equation, under certain integrability assumptions on the potential. Let

1 V ∈ L (R) (4.172) and study the operator

H = H0 + V, (4.173) 2 acting on L (R). Note that we allow for V to be complex-valued, meaning that the operator in Equation (4.173) in general is not, and will not be assumed to be, symmetric. Instead, we follow the techniques of [Abr01] to obtain a general upper bound on the magnitude of the eigenvalues of H. The main result of this section is the following.

Theorem 14. Let V be as in Equation (4.172). Any eigenvalue λ of H with λ 6∈ σ(H0) = [2, ∞) then satises

2 λ − 1 2 4 kV k 1 ≥ , (4.174) L (R) 2 1 ω λ q λ q λ + ln + + 1 − 1 4τ π 2 2 2 where the square roots are principal.

Remark 9. The right hand side of Equation (4.174) is plotted in Figure 4.1, for complex and real λ ∈ C r [2, ∞), respectively. Note that the bound is void for λ = −2. It also implies that eigenvalues to the operator

Hα := H0 + αV, α ∈ R, (4.175) can only cluster around the point λ = −2 or lie on [2, ∞), as α → 0.

2 Proof. Any eigenvalue λ with eigenfunction ψ ∈ L (R) satises

(H0 − λ) ψ = −V ψ, (4.176) i.e. V −1 1/2 ϕ = − (H0 − λ) |V | ϕ, (4.177) |V |1/2 44 CHAPTER 4. OPERATORS FROM CONFORMAL FIELD THEORY

0.4

1.5 0.35

0.3

0.5

Lower bound 0.25 0 10 0 −5 0 5 10 −10 0.2 Im(λ) Re(λ)

Lower bound 0.15

4000

0.1 2000

0 0.05 Lower bound 1000 500 1000 500 0 0 −500 −500 −1000 −1000 0 −10 −8 −6 −4 −2 0 2 Im(λ) Re(λ) λ (a) The right hand side of Equation (4.174) plot- (b) The right hand side of Equation (4.174) plotted for complex λ. ted for real λ. Figure 4.1: Eigenvalue-bounding functions in Theorem 14. for ϕ := V ψ. We note that ϕ ∈ L2( ), since |V |1/2 R

Z 2 Z V 2 2 ψ dx = |V | |ψ| dx ≤ kV k 1 kψk ∞ , (4.178) 1/2 L (R) L (R) R |V | R

∞ which is nite, as ψ ∈ L (R) by Remark 1. Because of this, we will study

V −1 1/2 T := − (H0 − λ) |V | , (4.179) |V |1/2

2 as an operator with domain L (R). In the following, we will show that T also maps 2 2 continuously into L (R). In fact, T is even Hilbert-Schmidt on L (R), for λ ∈ ρ(H0). We now introduce the parametrization πik , where corre- λ = 2 cosh ω k ∈ R sponds to a λ ∈ σ(H0) and 0 < Im(k) ≤ |ω| to a λ ∈ ρ(H0). In the latter case, the resolvent of H0 was calculated in [Fad15] and in Section 4.2 to be an integral operator with kernel R0(x − y; λ) for

ω e−2πikx e2πikx R (x; λ) := + . (4.180) 0 πik 1 − e−4πiωx 1 − e4πiωx sinh ω We can then write

Z V (T ϕ)(x) = − (x)R (x − y; λ) |V |1/2 (y)ϕ(y)dy, (4.181) 1/2 0 R |V | which will be used to derive the desired inequality. To proceed, we will bound

R0(x; λ) uniformly in x. 4.3. EIGENVALUE ESTIMATES FOR GENERAL POTENTIALS 45

Lemma 4. Write πik , for . We then have the bound λ = 2 cosh ω 0 < Im(k) ≤ |ω|

2 ! 2 2 |k| |ω| |R0(x; λ)| ≤ 1 + . (4.182) |ω|2 πik 2 sinh ω

Proof of Lemma 4. Note that R0(x; λ) is symmetric in x, so we can without loss of generality assume x ≥ 0. A straightforward calculation shows that

2 sinh πik cosh2 π( 1 − 2Im(k))x − cos2 (2πRe(k)x) ω 2 b (4.183) 2 |R0(x; λ)| = 2 πx |ω| sinh b cosh2 πx − cos2 (2πRe(k)x) b (4.184) ≤ 2 πx sinh b 2 1 − cos (2πRe(k)x) (4.185) = 1 + 2 πx sinh b 1 − cos2 (2πRe(k)x) ≤ 1 + =: f(x, k), (4.186) πx 2 b where the rst inequality used that k has positive imaginary part and the last that x ≤ sinh (x), which is obtained by dierentiation four times. f(x, k) is smooth and tends to zero as x → ∞. The end-point x = 0 has zero slope, and any zeros xn of

df(x, k) π2x πRe(k) sin (4πRe(k)x) x − 1 − cos2 (2πRe(k)x) = 2 (4.187) dx b2 πx 4 b satisfy

πRe(k) sin (4πRe(k)xn) xn 2 2 sin (4πRe(k)xn) (4.188) f(xn) = 1 + 2 = 1 + 4b Re(k) πxn 4πRe(k)xn b 2 k ≤ 1 + . (4.189) ω

Inserting this into Equation (4.186) concludes the proof of the lemma.

Returning to Equation (4.181), we x a λ ∈ ρ(H0) and note that T is then Hilbert-Schmidt with respect to the Lebesgue measure, since

Z V 2 kT k2 = (x)R (x − y; λ) |V |1/2 (y) dxdy (4.190) I2(X) 1/2 0 R2 |V | 2 ! 2 |k| |ω| 2 ≤ 1 + kV k 1 , (4.191) |ω|2 πik 2 L (R) sinh ω 46 CHAPTER 4. OPERATORS FROM CONFORMAL FIELD THEORY which is nite, by our assumptions on V . Now, this implies that

2 ! 2 |k| |ω| 2 2 2 1 + kV k 1 ≥ kT k ≥ kT k 2 (4.192) |ω|2 πik 2 L (R) I2(X) L (R) sinh ω !2 kT ϕk 2 ≥ L (R) = 1, (4.193) kϕk 2 L (R) where we used Equation (4.177) in the last step. Rewriting this using r r ! ω λ λ λ k = ln + + 1 − 1 (4.194) πi 2 2 2 yields the desired bound.

Self-adjoint operators This section treats some general techniques for the discretized Schrödinger equation with a real-valued potential V . We use the Birman-Schwinger principle to derive a general bound on the number of eigenvalues less than some pre-designated Λ ∈ R, as a function of the potential V . For potentials with V → ∞ as |x| → ∞, this in particular implies that the discrete spectrum has no accumulation points, so the essential spectrum equals the continuous one. We then bound also higher moments of eigenvalues. Let now as above −1 H := U + U + V = H0 + V, (4.195) for some real-valued potential V . The eigenvalues of H less than Λ, which we denote by N(Λ,H), are the negative eigenvalues of

H0 := U + U −1 + (V − Λ). (4.196)

By the min-max principle, the eigenvalues of H0 are all more positive than the eigenvalues of −1 H− := U + U + (V − Λ)−, (4.197) meaning that we have the triplet

0 N(Λ,H) = N(0,H ) ≤ N(0,H−). (4.198) We next proceed to bound this last term by using a Birman-Schwinger argument from [Bir61, Sch61]. This gives the following two main results.

Proposition 4. Let V (x) be essentially bounded from below, and take Λ ∈ R. The eigenvalue counting function then satises

Z (V (x) − Λ) (V (y) − Λ) 1 − − (4.199) N(Λ,H) ≤ dxdy. 16τ 2 2 π(x−y) R cosh 2b 4.3. EIGENVALUE ESTIMATES FOR GENERAL POTENTIALS 47

Using Young's inequality for the last integral, we also obtain a uniform bound C πx p,q,r;1 N(Λ,H) ≤ cosh · (V − Λ) p r − L ( ) 16τ 2b L (R) R (4.200)

· (V − Λ) q , − L (R) which we will use in a variation of an Aizenman-Lieb argument [Aiz78] to obtain bounds on moments of eigenvalues to H. This results in the following corollary.

Corollary 6. Let V (x) be essentially bounded from below, and let {λn} be eigenvalues to H. If γ ∈ R+, then ∞ X γΓ(γ)Γ(3)D(b) Z (λ − Λ)γ ≤ (V (x) − Λ)γ+2 dx, (4.201) n − 16τΓ(γ + 3) − n=1 R where

2 −1 D(b) := min D(p, b) · |{x : 0 < Λ − V (x)}| p (4.202) p∈[1,2] 2(p−1) 1 s 2−p ! p p p p0 p0p p Z dx := min (4.203) 0 1 p p∈[1,2] p0 2 2(2 − p) p−1 πx p R cosh 2b 2 −1 · |{x : 0 < Λ − V (x)}| p (4.204) is a constant, which can be bounded by 4b . D(b) ≤ min(|{x : 0 < Λ − V (x)}| , π ) Remark 10. We can numerically compute the constants D(p, b) and D(b) in Corol- lary 6. The result is shown in Figure 4.2. Before turning to the proofs of these claims, we rst list some of their con- sequences. Firstly, since the left hand side of Equation (4.199) is integer-valued, 2 Proposition 4 implies that potentials suciently small in L (R) have no negative eigenvalues. This contrasts with the continuous Schrödinger equation in one dimension, where any potential will have at least one negative eigenvalue, independent of its size. Proposition 4 also implies parts of the results in Theorem 12, as in the following corollary.

Corollary 7. If V (x) → +∞ as x → ±∞, then N(Λ,H) is nite, so the discrete spectrum of H has nite multiplicity and no accumulation points. In particular, the essential spectrum equals the continuous one. Proof of Corollary 7. Under these assumptions, the integral in the right hand side of Equation (4.199) is nite for every Λ, so the conclusion follows. Remark 11. Note that the right hand side in Equation (4.199) tends to +∞ as ω0 tends to zero in the limit of a continuous Schrödinger equation. The bound of the Proposition then becomes void, reecting the fact that the resolvent changes form in the limit ω0 → 0. 48 CHAPTER 4. OPERATORS FROM CONFORMAL FIELD THEORY

D(b) 1.2 4b/π 1

14 1

10 0.8

6 D(b) 0.6 D(p,b)

2 0.4

0 2

1.8 10 0.2 1.6 8 1.4 6 4 1.2 2 0 1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 p b b

(a) D(p, b). (b) D(b) for |{x : 0 < Λ − V (x)}| = 1. Figure 4.2: Numerical calculations of constants in Corollary 6.

Example 7. We take πix , and parametrize πik . Using V (x) = 2 cosh ω Λ = cosh ω Corollary 6 with γ = 0, we obtain Z D(b) 2 (4.205) N(Λ,H) ≤ (V − Λ)− dx 16τ R ( 3ω sinh(2 πik ) 2k cosh2 πik + k − ω , Λ ≥ 2, = ω 2πik (4.206) 0, Λ < 2.

For Λ → ∞, this implies that the bound on N(Λ,H) behaves as Λ2 ln(Λ), which is to be compared to the actual behaviour ln(Λ)2, obtained in Example 6.

Proof of Proposition 4. Put P = (V − Λ)− and take

1/2 −1 1/2 ∗ Kλ := P (H0 + λ) P = B B, (4.207)

1/2 −1/2 for B = P (H0 + λ) . Since V was assumed to be essentially bounded from ∞ −1 below, P is in L (R). Moreover, for λ 6∈ −σ(H0) = (−∞, −2], (H0 + λ) in Equation (4.180) satises

−2πIm(k)x |R0(x − y; λ)| ≤ Ce , (4.208)

p −1 which is in L (R) for all 1 ≤ p ≤ ∞. Schur's lemma then implies that (H0 +λ) is a 2 ∞ bounded operator from D(H) to L (R), meaning that also Kλ is, since P ∈ L (R). 2 Kλ is then a bounded, self-adjoint operator on L (R). We next show the duality result that eigenfunctions with eigenvalue λ to H correspond to eigenfunctions with eigenvalue 1 to Kλ. If we have a ψ ∈ D(H) with Hψ = −λψ, for some λ ≤ Λ, then

(H0 − P )ψ = −λψ, (4.209) 4.3. EIGENVALUE ESTIMATES FOR GENERAL POTENTIALS 49 i.e. 1/2 1/2 −1 1/2 1/2 P ψ = P (H0 + λ) P P ψ, (4.210) 1/2 2 or Kλϕ = ϕ, for ϕ = P ψ ∈ L (R). This shows that the eigenvalue λ to H corresponds to the eigenvalue 1 of Kλ 2 acting on L (R). Next, we claim that the eigenvalues of Kλ are decreasing functions of λ. To see this, note that in the momentum representation, we have

−1 −1 ψ(p) ψ(p) F(H0 + λ) F ψ (p) = ≤ (4.211) πip πip 0 2 cosh ω + λ 2 cosh ω + λ 0 −1 −1 = F(H0 + λ ) F ψ (p), (4.212) for λ0 ≤ λ. It follows that 1/2 −1 1/2 (4.213) hψ, KλψiL2(R) = hP ψ, (H0 + λ) P ψiL2(R) −1 (4.214) = hϕ, (H0 + λ) ϕiL2(R), for ϕ = P 1/2ψ. Since the Fourier-transform is unitary, we obtain

−1 −1 −1 hψ, K ψi 2 = hϕ, (H + λ) ϕi 2 = hϕ, F(H + λ) F ϕi 2 (4.215) λ L (R) 0 L (R) b 0 b L (R) 0 −1 −1 ≤ hϕ, F(H + λ ) F ϕi 2 = hψ, K 0 ψi 2 . (4.216) b 0 b L (R) λ L (R)

By the min-max principle, the eigenvalues of Kλ must then all decrease with λ. It follows that N(0,H−) is not greater than the number of eigenvalues of K0 that are greater than or equal to one. If we denote these by λj ≥ 1, then certainly X N(0,H ) ≤ λ2 ≤ kK k2 . (4.217) − j 0 I2(X) j≥1

Since the operator 1/2 −1 1/2 is an integral operator, we can calculate the K0 = P H0 P last norm to obtain Z 2 N(0,H−) ≤ P (x)P (y) |R0(x − y; 0)| dxdy. (4.218) R2 Equation (4.183) from the calculations in Lemma 4, give

2 2 1 2 2 |ω| cosh π( b − 2Im(k))x − cos (2πRe(k)x) |R0(x − y; λ)| = , (4.219) πik 2 2 πx sinh ω sinh b where we use the paremetrization πik , for . then λ = 2 cosh ω 0 ≤ Im(k) ≤ ω λ = 0 corresponds to ω , so k = 2

2 1 (4.220) |R0(x − y; 0)| = 2 πx . 16τ cosh 2b Inserting this into Equation (4.218) concludes the proof. 50 CHAPTER 4. OPERATORS FROM CONFORMAL FIELD THEORY

Proof of Corollary 6. By the min-max principle, we can without loss of generality assume that Λ−V (x) ≥ 0. We also introduce an auxiliary number µ ∈ R+ and note that the number of eigenvalues of H −Λ less than −µ equals the number of negative eigenvalues of H − Λ + µ. Aiming to set µ = 0, we sum over eigenvalues less than −µ and write Z X γ γ (4.221) (λn − Λ)− = − µ dN(H − Λ, −µ) + Λ−λn≤−µ R Z = γ µγ−1N(H − Λ, −µ)dµ, (4.222) R+ after integrating by parts. We now use the bound from Equation (4.200), with , p . This results in the upper bound 1 ≤ p = q ≤ 2 r = 2(p−1)

2 Z Z p X γ γD(p, b) γ−1 p (4.223) (λn − Λ)− ≤ µ (V (x) − Λ + µ)− dx dµ, 16τ + Λ−λn≤−µ R R where πx D(p, b) = Cp,p,r;1 cosh (4.224) r 2b L (R) is a constant. We put E(µ) = |{x : V (x) < Λ − µ}| and use Jensen's inequality to obtain the bound

γD(p, b) Z 2 Z X γ γ−1 p −1 2 (λn − Λ)− ≤ µ E(µ) (V (x) − Λ + µ)− dxdµ. 16τ + Λ−λn≤−µ R R (4.225) Changing from to the variable µ , we can rewrite this bound as µ y = Λ−V

Z 1 Z γD(p, b) 2 −1 γ−1 p γ 2 (4.226) y E [(V − Λ)y] (Λ − V ) [(V (x) − Λ)(1 − y)]− dxdy 16τ 0 R Z 1 Z 2 −1 γD(p, b) p γ−1 2 γ+2 (4.227) ≤ E(0) y (1 − y)+dy (Λ − V (x)) dx 16τ 0 R 2 −1 γΓ(γ)Γ(3)D(p, b)E(0) p Z = (Λ − V )γ+2dx. (4.228) 16τΓ(γ + 3) R

2 −1 It therefore only remains to give a bound on D(b) := minp∈[1,2] D(p, b)E(0) p . Taking , , we can explicitly compute , 4b , p = 1 p = 2 D(1, b) = E(0) D(2, b) = π respectively, which nishes the proof. Bibliography

[Abr01] Abramov, A.A. and Aslanyan, A. and Davies, E.B., Bounds on complex eigenvalues and resonances, Journal of Physics A: Mathematical and Gen- eral 34 (2001), no. 1, 57. [Aiz78] Aizenman, M. and Lieb, E.H., On semi-classical bounds for eigenvalues of Schrödinger operators, Physics Letters A 66 (1978), no. 6, 427429. [Ben00] Benguria, R. and Loss, M., A simple proof of a theorem of Laptev and Weidl., Math. Res. Lett. 7 (2000), no. 2, 195203. [Bir61] Birman, M.S., On the spectrum of singular boundary-value problems, Matematicheskii Sbornik 97 (1961), no. 2, 125174. [Blu09] Blumenhagen, R. and Plauschinn, E., Introduction to Conformal Field Theoy: With Applications to String Theory, Springer, Berlin, 2009. [Bre83] Brezis, H., Analyse fonctionnelle, vol. 33, Masson, Paris, 1983. [Bri88] Brislawn, C., Kernels of trace class operators, Proceedings of the American Mathematical Society 104 (1988), no. 4, 11811190. [Con88] Constantin, P. and Foias, C. and Temam, R., On the dimension of the at- tractors in two-dimensional turbulence, Physica D: Nonlinear Phenomena 30 (1988), no. 3, 284296. [Cwi77] Cwikel, M., Weak type estimates for singular values and the number of bound states of Schrödinger operators., Ann. Math. 106 (1977), no. 1, 93 100.

[Der14] Derkachev, S.E. and Faddeev, L.D., 3j-symbol for the modular double SLq(2, R) revisited, Journal of Physics: Conference Series, vol. 532, IOP Publishing, 2014, p. 012005.

51 52 BIBLIOGRAPHY

[Dys67] Dyson, F.J. and Lenard, A., Stability of matter. I, Journal of mathematical physics 8 (1967), no. 3, 423434.

[Exn14] Exner, P. and Laptev, A. and Usman, M., On Some Sharp Spectral Inequal- ities for Schrödinger Operators on Semiaxis, Communications in Mathe- matical Physics 26 (2014), no. 2, 531541.

[Fad86] Faddeev, L.D. and Takhtajan, L.A., Liouville model on the lattice, Field Theory, Quantum Gravity and Strings II : Proceedings of a Seminar Series Held at DAPHE, Observatoire de Meudon, and LPTHE, Université Pierre et Maire Curie, Paris, Between October 1985 and October 1986, Springer, Berlin, 1986, pp. 166179.

[Fad01] Faddeev, L.D. and Kashaev, R.M. and Volkov, A.Y., Strongly Coupled Quantum Discrete Liouville Theory I: Algebraic Approach and Duality, Communications in Mathematical Physics 219 (2001), no. 1, 199219.

[Fad15] Faddeev, L.D. and Takhtajan, L.A., On the spectral theory of one functional-dierence operator from conformal eld theory, Izvestiya RAN: Ser. Mat. 79 (2015), no. 2, 181204, English translation in Izvestiya: Mathematics 79 (2015), no. 2, 388-410.

[Fis66] Fisher, M.E. and Ruelle, D., The stability of many-particle systems, Jour- nal of Mathematical Physics 7 (1966), no. 2, 260270.

[Fra07] Frank, R.L. and Lieb, E.H. and Seiringer, R., Stability of relativistic matter with magnetic elds for nuclear charges up to the critical value, Commu- nications in Mathematical Physics 275 (2007), no. 2, 479489.

[Fri34] Friedrichs, K., Spektraltheorie halbbeschränkter Operatoren und Anwen- dung auf die Spektralzerlegung von Dierentialoperatoren, Mathematische Annalen 109 (1934), no. 1, 465487.

[Fri82] Friedman, A., Foundations of Modern Analysis, Courier Dover Publica- tions, New York, 1982.

[Gol65] Golden, S., Lower bounds for the Helmholtz function, Physical Review 137 (1965), no. 4B, 11271128.

[Gra09] Grafakos, L., Modern fourier analysis, vol. 250, Springer, New York, 2009.

[Gra14] Grassi, A. and Hatsuda, Y. and Marino, M., Topological Strings from Quantum Mechanics, arXiv preprint arXiv:1410.3382 (2014).

[Hun02] Hundertmark, D. and Lieb, E.H. and Thomas, L.E., A sharp bound for an eigenvalue moment of the one-dimensional Schrödinger operator, Inequal- ities, Springer, Berlin Heidelberg, 2002, pp. 329341. BIBLIOGRAPHY 53

[Kas01a] Kashaev, R., The quantum dilogarithm and Dehn twists in quantum Te- ichmüller theory, Integrable structures of exactly solvable two-dimensional models of quantum eld theory, Springer, Netherlands, 2001, pp. 211221.

[Kas01b] Kashaev, R.M., On the spectrum of Dehn twists in quantum Teichmüller theory, Physics and Combinatorics: Proceedings of the Nagoya 2000 In- ternational Workshop, Nagoya University, 21 26 August 2000, 2001, pp. 6381.

[Lap97] Laptev, A., Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces, Journal of Functional Analysis 151 (1997), no. 2, 531 545.

[Lap00] Laptev, A. and Weidl, T., Sharp Lieb-Thirring inequalities in high dimen- sions, Acta Mathematica 184 (2000), no. 1, 87111.

[Lie76] Lieb, E.H. and Thirring, W.E., Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann, Princeton University Press, Princeton, 1976.

[Lie88] Lieb, E.H. and Yau, H.-T., The stability and instability of relativistic matter, Communications in Mathematical Physics 118 (1988), no. 2, 177213.

[Lie97a] Lieb, E.H. and Loss, M. and Solovej, J.P., Stability of matter in magnetic elds, The Stability of Matter: From Atoms to Stars, Springer, Berlin Heidelberg, 1997, pp. 432436.

[Lie97b] Lieb, E.H. and Siedentop, H. and Solovej, J.P., Stability and instability of relativistic electrons in classical electromagnetic elds, Journal of Statisti- cal Physics 89 (1997), no. 1-2, 3759.

[Lie01a] Lieb, E.H., Bound on the maximum negative ionization of atoms and molecules, The Stability of Matter: From Atoms to Stars, Springer, Berlin Heidelberg, 2001, pp. 91101.

[Lie01b] , The number of bound states of one-body Schrödinger operators and the Weyl problem, The Stability of Matter: From Atoms to Stars, Springer, Berlin Heidelberg, 2001, pp. 245256.

[Lie01c] Lieb, E.H. and Loss, M., Analysis, vol. 14, American Mathematical Soci- ety, Providence, RI, 2001.

[Lie02] Lieb, E.H., On characteristic exponents in turbulence, Inequalities, Springer, Berlin Heidelberg, 2002, pp. 305312.

[Lie09] Lieb, E.H. and Seiringer, R., The stability of matter in quantum mechanics, Cambridge University Press, Cambridge, 2009. 54 BIBLIOGRAPHY

[Maz85] Maz'ya, V., Sobolev Spaces with applications to elliptic partial dierential equations, Springer, Berlin, 1985.

[Mic15] Mickelin, O., LiebThirring inequalities for generalized magnetic elds, Bulletin of Mathematical Sciences (2015), 114.

[Nam12] Nam, P.T., New bounds on the maximum ionization of atoms, Communi- cations in Mathematical Physics 312 (2012), no. 2, 427445.

[Pon01] Ponsot, B. and Teschner, J., ClebschGordan and RacahWigner Coe- cients for a Continuous Series of Representations of Uq(Sl(2, R)), Com- munications in Mathematical Physics 224 (2001), no. 3, 613655.

[Ree72] Reed, M. and Simon, B., Methods of modern mathematical physics I: Func- tional analysis, vol. 2, Academic press, New York, 1972.

[Ree75] , Methods of modern mathematical physics II: Fourier analysis, self- adjointness, vol. 2, Academic Press, New York, 1975.

[Ree78] , Methods of modern mathematical physics IV: Analysis of Opera- tors, vol. 4, Academic Press, New York, 1978.

[Ros04] Rosenberg, J., A selective history of the Stone-von Neumann theorem, Con- temporary Mathematics: Operator Algebras, Quantization, and Noncom- mutative Geometry, vol. 365, American Mathematical Society, Providence, RI, 2004, pp. 331354.

[Roz76] Rozenbljum, G.V., Distribution of the discrete spectrum of singular dierential operators, Izv. Vyssh. Uchebn. Zaved. Mat. 20 (1976), no. 1, 7586, English translation in Math. Izv. VUZ 20 (1976), no. 1, 6371.

[Rui97] Ruijsenaars, S.N.M., First order analytic dierence equations and integrable quantum systems, Journal of Mathematical Physics 38 (1997), no. 2, 10691146.

[Sch61] Schwinger, J., On the bound states of a given potential, Proceedings of the National Academy of Sciences of the United States of America 47 (1961), no. 1, 122129.

[Sei10] Seiringer, R., Inequalities for Schrödinger operators and applications to the stability of matter problem, Contemporary Mathematics: Entropy and the Quantum, vol. 529, American Mathematical Society, Providence, RI, 2010, pp. 5372.

[Sim79] Simon, B., Functional integration and quantum physics, vol. 86, Academic press, New York, 1979.

[Sim09] , Schrödinger operators with purely discrete spectrum, Methods of Functional Analysis and Topology 15 (2009), no. 1, 6166. BIBLIOGRAPHY 55

[Sol06] Solovej, J.P, Stability of Matter, Encyclopedia of Mathematical Physics, vol. 5, Elsevier Science, Amsterdam, 2006, pp. 814.

[Str08] Struwe, M., Funktionalanalysis I und II, https://people.math.ethz. ch/~struwe/Skripten/FA-I-II-26-8-08.pdf, 2008, [Online; accessed 7- April-2015].

[Tak08] Takhtajan, L.A., Quantum mechanics for mathematicians, vol. 95, Amer- ican Mathematical Society, Providence, RI, 2008.

[Tho65] Thompson, C.J., Inequality with applications in statistical mechanics, Journal of Mathematical Physics 6 (1965), no. 11, 18121813.

[Vol05] Volkov, A.Y., Noncommutative hypergeometry, Communications in Math- ematical Physics 258 (2005), no. 2, 257273.

[Wei96] Weidl, T., On the Lieb-Thirring constants Lγ,1 for γ ≥ 1/2, Communica- tions in Mathematical Physics 178 (1996), no. 1, 135146.

[Wey11] Weyl, H., Über die asymptotische Verteilung der Eigenwerte, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch- Physikalische Klasse 1911 (1911), no. 1911, 110117.

[Zam96] Zamolodchikov, A.B. and Zamolodchikov, Al. B., Conformal bootstrap in Liouville eld theory, Nuclear Physics B 477 (1996), no. 2, 577605.

Paper A. LiebThirring inequalities for generalized magnetic elds

TRITA -MAT-E 2015:17 ISRN -KTH/MAT/E--15/17--SE