A spectral theory of linear operators on rigged Hilbert spaces under analyticity conditions II: applications to Schr\"odinger operators

TL;DR

This paper develops a spectral theory for linear operators on rigged Hilbert spaces and applies it to Schrödinger operators, providing a unified approach to resonances without spectral deformation, and introduces a new formulation of the Evans function.

Contribution

It introduces a novel spectral theory framework on rigged Hilbert spaces and applies it to Schrödinger operators, connecting generalized eigenvalues with a new holomorphic function.

Findings

Unified approach to resonances for different potentials

Construction of a holomorphic function characterizing eigenvalues

New formulation and analytic continuation of the Evans function

Abstract

A spectral theory of linear operators on a rigged Hilbert space is applied to Schr\"odinger operators with exponentially decaying potentials and dilation analytic potentials. The theory of rigged Hilbert spaces provides a unified approach to resonances (generalized eigenvalues) for both classes of potentials without using any spectral deformation techniques. Generalized eigenvalues for one dimensional Schr\"odinger operators (ordinary differential operators) are investigated in detail. A certain holomorphic function $\mathbb{D}(\lambda)$ is constructed so that $\mathbb{D}(\lambda) = 0$ if and only if $\lambda $ is a generalized eigenvalue. It is proved that $\mathbb{D}(\lambda)$ is equivalent to the analytic continuation of the Evans function. In particular, a new formulation of the Evans function and its analytic continuation is given.