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Scattering Theory Wi4211 Advanced Topics in Analysis Scattering Theory wi4211 Advanced Topics in Analysis Erik Koelink Spring 2006 ii Contents 1 Introduction 1 1.1 Scattering theory . 1 1.2 Inverse scattering method . 2 1.3 Overview . 3 2 Schr¨odinger operators and their spectrum 5 d2 2.1 The operator − dx2 ................................. 5 d2 2.2 The Schr¨odingeroperator − dx2 + q ........................ 7 2.3 The essential spectrum of Schr¨odinger operators . 12 2.4 Bound states and discrete spectrum . 15 2.5 Explicit examples of potentials . 18 2.5.1 Indicator functions as potential . 18 2.5.2 cosh−2-potential . 19 2.5.3 Exponential potential . 23 3 Scattering and wave operators 27 3.1 Time-dependent Schr¨odinger equation . 27 3.2 Scattering and wave operators . 28 3.3 Domains of the wave operators for the Schr¨odingeroperator . 35 3.4 Completeness . 37 4 Schr¨odinger operators and scattering data 47 4.1 Jost solutions . 47 4.2 Scattering data: transmission and reflection coefficients . 57 4.3 Fourier transform properties of the Jost solutions . 65 4.4 Gelfand-Levitan-Marchenko integral equation . 70 4.5 Reflectionless potentials . 77 5 Inverse scattering method and the Korteweg-de Vries equation 83 5.1 The Korteweg-de Vries equation and some solutions . 83 5.2 The KdV equation related to the Schr¨odinger operator . 89 5.3 Lax pairs . 91 5.4 Time evolution of the spectral data . 95 iii iv 5.5 Pure soliton solutions to the KdV-equation . 97 6 Preliminaries and results from functional analysis 103 6.1 Hilbert spaces . 103 6.2 Operators . 106 6.3 Fourier transform and Sobolev spaces . 108 6.4 Spectral theorem . 112 6.5 Spectrum and essential spectrum for self-adjoint operators . 114 6.6 Absolutely continuous and singular spectrum . 119 Bibliography 123 Index 125 Chapter 1 Introduction According to Reed and Simon [9], scattering theory is the study of an interacting system on a time and/or distance scale which is large compared to the scale of the actual interaction. This is a natural phenomenon occuring in several branches of physics; optics (think of the blue sky), acoustics, x-ray, sonar, particle physics,. In this course we focus on the mathematical aspects of scattering theory, and on an important application in non-linear partial differential equations. 1.1 Scattering theory As an example motivating the first chapters we consider the following situation occuring in quantum mechanics. Consider a particle of mass m moving in three-dimensional space R3 according to a potential V (x, t), x ∈ R3 the spatial coordinate and time t ∈ R. In quantum R 2 mechanics this is modelled by a wave function ψ(x, t) satisfying 3 |ψ(x, t)| dx = 1 for all R time t, and the wave function is interpreted as a probability distribution for each time t. This means that for each time t, the probability that the particle is in the set A ⊂ R3 is given by R 2 A |ψ(x, t)| dx. Similarly, the probability distribution of the momentum for this particle is given by |ψˆ(p, t)|2, where Z ˆ 1 −ip·x ψ(p, t) = √ 3 e ψ(x, t) dx 3 2π R is the Fourier transform of the wave function with respect to spatial variable x. (Here we have scaled Planck’s constant ~ to 1.) Using the fact that the Fourier transform interchanges differentiation and multiplication the expected value for the momentum can be expressed in terms of a differential operator. The kinetic energy, corresponding to |p|2/2m, at time t of −1 the particle can be expressed as 2m h∆ψ, ψi, where we take the inner product corresponding to the Hilbert space L2(R3) and ∆ is the three-dimensional Laplacian (i.e. with respect to the spatial coordinates x). 1 2 Chapter 1: Introduction The potential energy at time t of the particle is described by hV ψ, ψi, so that (total) energy of the particle at time t can be written as −1 E = hHψ, ψi,H = ∆ + V, 2m where we have suppressed the time-dependence. The operator H is known as the energy operator, or as Hamiltonian, or as Schr¨odingeroperator. In case the potential V is independent of time and suitably localised in the spatial coordinate, we can view H as a perturbation of −1 the corresponding free operator H0 = 2m ∆. This can be interpreted that we have a ‘free’ particle scattered by the (time-independent) potential V . The free operator H0 is a well- known operator, and we can ask how its properties transfer to the perturbed operator H. Of course, this will depend on the conditions imposed on the potential V . In Chapter 2 we study this situation in greater detail for the case the spatial dimension is 1, but the general perturbation techniques apply in more general situations as well. In general the potentials for which these results apply are called short-range potentials. In quantum mechanics the time evolution of the wave function is determined by the time- dependent Schr¨odinger equation ∂ i ψ(x, t) = Hψ(x, t). (1.1.1) ∂t We want to consider solutions that behave as solutions to the corresponding free time- dependent Schr¨odinger equation, i.e. (1.1.1) with H replaced by H0, for time to ∞ or −∞. In case of the spatial dimension being 1 and for a time-independent potential V , we study this situation more closely in Chapter 2. We do this for more general (possibly unbounded) self-adjoint operators acting a Hilbert space. 1 Let us assume that m = 2 , then we can write the solution to the free time-dependent Schr¨odinger equation as Z ψ(x, t) = F (p) eip·xe−i|p|2t2 dp, 3 R where F (p) denotes the distribution of the momenta at time t = 0 (up to a constant). This ip·x follows easily since the exponentials e are eigenfunctions of H0 = −∆ for the eigenvalue |p|2. For the general case the solution is given by Z −i|p|2t2 ψ(x, t) = F (p) ψp(x) e dp, 3 R 2 ip·x where Hψp = |p| ψp and where we can expect ψp(x) ∼ e if the potential V is sufficiently ‘nice’. We study this situation more closely in Chapter 4 in case the spatial dimension is one. 1.2 Inverse scattering method As indicated in Section 4.2 we give an explicit relation between the potential q in the Schr¨o- d2 dinger operator − dx2 + q and its scattering data consisting of the reflection and transition Chapter 1: Introduction 3 coefficient R and T in Chapter 4. To be specific, in Section 4.2 we discuss the transition q 7→ {R, T }, which is called the direct problem, and in Section 4.4 we discuss the transition {R, T } 7→ q, the inverse problem. In case we take a time-dependent potential q(x, t), the scattering data also becomes time-dependent. It has been shown that certain non-linear partial differential equations (with respect to t and x) for q imply that the corresponding time evolution of the scattering data R and T is linear! Or, scattering can be used to linearise some non-linear partial differential equations. The basic, and most famous, example is the Korteweg-de Vries equation, KdV-equation for short, qt(x, t) − 6 q(x, t) qx(x, t) + qxxx(x, t) = 0, which was introduced in 1894 in order to model the solitary waves encountered by James Scott Russell in 1834 while riding on horseback along the canal from Edinburgh to Glasgow. Although the Korteweg-de Vries paper “On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves”, Philosophical Magazine, 39 (1895), 422–443, is now probably the most cited paper by Dutch mathematicians, the KdV-equation lay dormant for a long time until it was rediscovered by Kruskal and Zabusky in 1965 for the Fermi-Pasta-Ulam problem on finite heat conductivity in solids. Kruskal and Zabusky performed numerical experiments, and found numerical evidence for the solitary waves as observed by Scott Russell, which were named “solitons” by Kruskal and Zabusky. The fundamental discovery made by Gardner, Greene, Kruskal and Miura in 1967 is that if q is a solution to the KdV-equation, then the spectrum of the Schr¨odingerwith time-dependent potential q is independent of time, and moreover, the time evolution for the scattering data, i.e. the reflection and transmission coefficient, is a simple linear differential equation. We discuss this method in Chapter 5. In particular, we discuss some of the soliton solutions. This gave way to a solution method, nowadays called the inverse scattering method, or inverse spectral method, for classes of non-linear partial differential equations. Amongst others, similar methods work for well-known partial differential equations such as the modified 2 KdV-equation qt − 6q qx + qxxx = 0, the sine-Gordon equation qxt = sin q, the non-linear 2 Schr¨odinger equation qt = qxx + |q| q, and many others. Nowadays, there are many families of partial differential equations that can be solved using similar ideas by realising them as conditions for isospectrality of certain linear problem. For the isospectrality we use the method of Lax pairs in Section 5.3. There have been several approaches and extensions to the KdV-equations, notably as integrable systems with infinitely many conserved Poisson-commuting quantities. We refer to [1], [8]. 1.3 Overview The contents of these lecture notes are as follows. In Chapter 6 we collect some results from functional analysis, especially on unbounded self-adjoint operators and the spectral theorem, Fourier analysis, especially related to Sobolev and Hardy spaces.
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