Scattering Theory Sean Harris October 2016 A thesis submitted for the degree of Bachelor of Philosophy (Honours) of the Australian National University Declaration The work in this thesis is my own except where otherwise stated. The first four chapters are largely expository, covering background material. The final two chapters contain mostly new theorems (to the best of my knowledge) proven by myself, with some help by others such as Professor Andrew Hassell. Sean Harris Acknowledgements Most of all I would like to thank my supervisor Professor Andrew Hassell, for developing my interest in this topic, always finding time to meet with me to discuss issues, and being generally great. I am also incredibly grateful to Dr Melissa Tacy for helping me draft this thesis while Andrew was absent. Thanks also goes to the many lecturers I have met leading up to this moment, for nourishing my interest in mathematics. My fellow Dungeon-dwelling honours students have also been incredibly helpful throughout the year. Special mention to Hugh McCarthy, for always being available to chat about any problems I was having (both mathematical and otherwise). A few more people to mention: my tutorial students, for giving me some reprieve from study; the staff at the ANU Food Co-op, for the great lunches (I wish I had discovered them earlier); and the many green envelopes and the single red envelope, for keeping me going. I guess I should thank my parents too. v Abstract Scattering theory studies the comparison between evolution obeying \free dynamics" and evolution obeying some \perturbed dynamics". The asymptotic nature of free and perturbed evolution are compared to determine properties of the perturbation. A brief introduction to scattering theory and inverse scattering problems is given in Chapter3, after covering some relevant analysis concepts and the construction of Laplacian and Dirichlet to Neumann operators in Chapters1 and2. The spectral duality result of [EP95] is as follows. Theorem ([EP95], Main Theorem). The following are equivalent: 2 1. −∆D has an M-fold degenerate eigenvalue k0 2. As k " k0, exactly M eigenphases θj(k) of SD(k) converge to π from below. In Chapter4 this result is explained both mathematically and intuitively, and the significance of the form of the second equivalent statement is examined. Possible generalisations of the main theorem of [EP95] to potential scattering are then inves- tigated in Chapter5. The Cayley transform generalisation, as explained in Section 5.3, suggests studying an object which is given the name transmission operator. The transmission operator displays qualities similar to the scattering matrix and so is investigated further in Chapter6. Many results about the spectrum of the transmission operator are proven under certain physically reasonable conditions on the considered potential for scattering. vii Introduction Our path to Scattering theory starts with the following problem: Suppose we wish to determine properties of a person's brain without intrusive testing. One possible way proceed would be to pass waves through the head, and observe the scattered waves which exit. We would like to know that this scattering information can return useful information, and so we will frame this as a mathematical problem. To find a starting point for such a theory, we may use the following thought experiment: Suppose n you stand at the edge of a large lake in R (this is mathematics, so the lake will of course be a ball). On top of this, suppose it is 2 a.m. and you are wearing your nicest shoes. In the middle of the lake, you can vaguely make out the shape of an object. Naturally you wish to determine properties of this object but you can't see it very well due to the time of day, and you don't want to ruin your shoes by swimming out to the object. All you can do is make splashes on the boundary of the lake, and measure the returning echoes (see Figure1). From this wave scattering information, would it be possible to gain any knowledge of the object? Figure 1: You stand on the edge of a lake and splash at the mysterious object. The grey solid line is the incoming and reflected wave, and the grey dotted line is the path the wave would have taken if there was no object. ix x The mathematical framework behind trying to solve such a problem is known as scattering theory. There are a few varieties of scattering theory, and the two we'll focus on are potential and obstacle scattering, which both fall under the heading of concrete scattering theory. Potential scattering can be thought of as modelling waves passing through localised inhomogeneity in a media (such as a change of refractive index), while obstacle scattering can be thought of as modelling waves travelling through space with some forbidden region (such as electromagnetic waves travelling through a region with a conductive chunk of metal in the way). Such a theory clearly has many real world applications, and it also has many interesting mathe- matical results! One such result is the spectral duality result of Eckmann and Pillet, given in their paper [EP95]. This spectral duality roughly states that an obstacle D appears \almost invisible" to some incoming radiation at frequencies approaching a given frequency k0, if and only if D can support standing waves of the same frequency k0. The aim of this thesis is to explain the result of Eckmann and Pillet and to attempt to generalise the result to potential scattering. To do this, we shall start by covering some basic analysis facts, using [RS80] and [Are15] as main recource. These will be used to construct the Laplacian - a key component of scattering theory, and the Dirichlet to Neumann operators - a key component of our attempts to generalise the result of [EP95]. The Laplacian is a differential operator which governs the mathematical understanding of many physical processes. To us, its significance will be in passing from physical waves travelling through a lake to mathematical waves travelling through wherever the reader believes mathematics to live. The Dirichlet to Neumann operators relate the boundary values of some function satisfying certain physical laws on a domain, to the flux of said function through the boundary. These operators will be used to come up with a \local problem" to match the potential scattering problem, in the same way the standing wave problem matches the scattering off D. After constructing these operators, we will explain how to pass from our physical problem of mystery objects in lakes and brain scans to a mathematical framework, known as scattering theory (following [Mel95]). This will leave us with enough knowledge to tackle the proof of Eckmann and Pillet's, of which we will cover the main ideas. Finally, in the last two chapters we shall develop some possible extensions of the spectral duality result to potential scattering. One of these extensions seems quite promising, and a few results concerning it are proven. The promising extension is given in terms of an operator, which we have named the \transmission operator". The transmission operator shows many similarities to the scattering matrix, a key component of scattering theory and the result of [EP95]. A proposed analogue of the main result of [EP95] related to the transmission operator is given in Section 5.3, as ConjectureB. A slightly modified form of this conjecture is proven for central potentials in Section 6.1, which is a new result of the author's. Many preliminary results towards a proof of ConjectureB are also proven as new results. All unreferenced results of Chapter6 are the author's work besides Lemma 6.10, which was joint work with Professor Andrew Hassell. Contents Acknowledgementsv Abstract vii Introduction ix Notation xiii 1 Background Analysis Concepts1 1.1 Unbounded Operators...................................1 1.2 Differentiability and Holomorphicity in Banach Spaces.................5 1.3 The Spectral Theorem...................................6 1.4 Forms............................................ 10 1.5 Groups and Semigroups.................................. 11 2 The Laplacian and Dirichlet to Neumann Operators 15 2.1 The Laplacian........................................ 15 2.1.1 The Laplacian in Spherical Coordinates..................... 17 2.2 The Spectral Theorem applied to ∆........................... 19 2.3 Dirichlet to Neumann Operators............................. 20 2.3.1 Dirichlet to Neumann Operator with respect to −∆.............. 21 2.3.2 Dirichlet to Neumann Operator with respect to −∆ + A ............ 22 3 Scattering Theory 25 3.1 Trivial Scattering...................................... 25 3.2 Non-Trivial Scattering................................... 27 3.3 Inverse Scattering Problem................................ 30 3.3.1 Potential Scattering Inverse Problem....................... 30 3.3.2 Obstacle Scattering Inverse Problems...................... 31 3.4 Abstract Scattering Theory................................ 32 4 Review of Spectral Duality for Planar Billiards 35 4.1 Spectral Duality - Easy Version.............................. 35 4.2 Difficulties with a Converse................................ 36 xi xii CONTENTS 4.3 Spectral Duality - Hard Version.............................. 37 4.3.1 Potential Theory.................................. 38 4.3.2 Relating Ak to SD(k)............................... 39 4.3.3 Proof of Theorem 4.5............................... 41 5 Generalisations of Spectral Duality to Potential Scattering 45 5.1 Dirichlet to Neumann Boundary Condition Generalisation..............