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Scattering theory for half-line Schrödinger operators: analytic and topological results

Inoue, Hideki 井上, 秀樹 名古屋大学

2020.10.19

概要

In 1949, N. Levinson pointed out a surprising relation between the number of bound states and the scattering part of a quantum system. Several works have been devoted to the generalization of this relation by many mathematicians and physicists. Various methods have also been used for the proof of this relation, e.g. the Jost function, the Green functions, the Strum-Liouville theorem, and the spectral shift functions. Relations of this type are now referred to as Levinson’s theorem.

A radically different approach to Levinson’s theorem is first proposed for one dimensional Schrödinger operators with a point interaction by J. Kellendonk and S. Richard. In this approach, the main analytical tools are the wave operators, which are central objects in mathematical scattering theory. For the last about ten years, it has been proved for several models that once recast in a C*-algebraic framework this relation can be understood as an index theorem for the wave operators. Resulting index theorems are called topological version of Levinson's theorem or shortly topological Levinson's theorem.

In this thesis, we present new analytical and topological results concerning Levinson's theorem for Schrödinger operators on the continuous and discrete half-line. This thesis is divided into four chapters, where the first chapter is regarded as an overview while the other three chapters are based on the published papers.

In the first chapter, we provide the historical background and the framework of our investigations. After introducing the models to be considered, we recall the usual formulation of Levinson’s theorem in terms of the Jost function. Then, we recall the abstract spectral and scattering theory as well as K-theory for C*-algebras. Finally, we discuss how to combine the analytic part with the algebraic part in applications.

In the second chapter, we consider Schrödinger operators with a real-valued bounded potential on the continuous half-line. A new perturbative estimate is established for the Jost solution of the corresponding Schrödinger equation under an optimal hypothesis on the decay of the potential, which ensures the finiteness of the number of bound states of the system. This estimate is used to prove the compactness of the remainder term in an explicit formula for the wave operators. Topological Levinson’s theorem is also provided with a detailed illustration. For such a result, it is the first time that the optimal decay is reached.

In the third chapter, we consider Schrödinger operators with several complex parameters on the continuous half-line. It has been shown that depending on several parameters, such operators possess either a finite number of complex eigenvalues, or an infinite one, but also some spectral singularities embedded in the continuous spectrum (exceptional situations). We first recall the spectral and the scattering theory for these operators. Then, new analytical results for the exceptional cases are provided. Topological Levinson’s theorem involving complex and infinitely many eigenvalues are also developed, and explanations why these results cannot be extended to the exceptional cases are provided.

In the fourth chapter, we consider Schrödinger operators on the discrete half-line. This system can be viewed as a finite difference approximation of the continuous system considered in the second chapter. Indeed, the results in this chapter is parallel to the second chapter: we deduce an explicit formula for the wave operators from their stationary expressions and prove a topological Levinson’s theorem as an application. We also discuss a relation between our usual C*-algebras and the famous Toeplitz algebra generated by the shift operator.

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