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Spectral and scattering theory for generalized Schrödinger operators

平良, 晃一 東京大学 DOI:10.15083/0002005442

2022.07.13

概要

In this thesis, we study spectral and scattering properties for generalized Schr¨odigner operators. In particular, we investigate essential self-adjointness and limiting absorption principle for some differential operators on Rd and Zd.

In Chapter 3, we show the essential self-adjointness and the limiting absorption principle for a d’Alembert operator on a Lorentzian space. Unlike the elliptic case, its proof is non-trivial. Moreover, we need geometric conditions even for operators on the Euclidean space with asymptotically constant coefficients.

In Chapter 4, we study the spectral properties of a repulsive Schr¨odinger operator. We give a micolocal proof for the classical result on its essential self-adjointness. A spectral property of its self-adjoint extensions is also studied.

In Chapter 5, we study the uniform bound of a Birman-Schwinger operator on a square lattice. For uniformly decaying potentials, we obtain the same bound as in the continuous setting. However, for non-uniformly decaying potential, our results are weaker than in the continuous setting.

In Chapter 6, we investigate Lp-mapping properties and the Carleman estimate of a Fourier multiplier operator and its resolvent. As an application, we prove existence and completeness of wave operators for a Dirac operator and a fractional Laplacian.

In Chapter 7, we address the precise asymptotic expansions and non-existence of resonant states for a discrete Schr¨odinger operator near its threshold energy.

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