Levinson’s Theorem: An Index Theorem in Scattering Theory

Abstract

A topological version of Levinson's theorem is presented. Its proof relies on a C * -algebraic framework which is introduced in detail. Various scattering systems are considered in this framework, and more coherent explanations for the corrections due to thresholds effects or for the regularization procedure are provided. Potential scattering, point interactions, Friedrichs model and Aharonov-Bohm operators are part of the examples which are presented. Every concepts from scattering theory or from K-theory are introduced from scratch.

Does Levinson’s theorem count complex eigenvalues?

“…In the following statement, we provide an alternative representation of the wave operator which is at the root of the algebraic framework presented in [9]. For that purpose, note first that for the special choice ( 1 2 , 0) the transformation F +# 1 2 ,0 ≡ F + 1 2 ,0 corresponds to e i π 4 F D with F D the usual Fourier sine transformation on R + .…”

“…Very often, this latter expression involves a regularization procedure, and in many cases some corrections must also be taken into account. Since the literature on the subject is very vast, we simply refer to the review papers [8,9] and to the references mentioned therein.…”

“…Nevertheless, these operators can be defined and their properties studied. Our approach for proving a Levinson's type theorem is based on the topological approach firstly introduced [6,7] and extensively reviewed in [9]. In this framework, Levinson's theorem corresponds to an index theorem in scattering theory, and the corrections come from the contribution of newly introduced operators living at threshold energies.…”

Schrödinger Wave Operators on the Discrete Half-Line

“…This new representation enables us to consider U , and therefore W − as pseudo-differential operators on L 2 (R). Note that the expression for the operator (U + 1)/2 in (3.4) in front of S − 1 is universal in the sense that it is independent of the potential, and such a phenomenon has been commonly observed for continuous systems (see [9] for more information).…”

“…Note that η is determined only up to modulo 2π. However, this is sufficient for our purpose since we are interested in an identity of the form 9) where N := #σ p (H) < ∞ is the number of bound states of H (see [12,Rem. 4.12]) and the correction terms ∆ ± = 0 if Ω(±1) = 0 and ∆ ± = 1/2 if Ω(±1) = 0.…”