Resonance Theory for Schroedinger Operators

TL;DR

This paper develops a general, weaker-condition theory for resonances in Schrödinger operators, addressing their decay rates, uniqueness, and relation to Green's functions through time-dependent methods.

Contribution

It introduces a novel, more flexible framework for analyzing resonances, including perturbations of threshold eigenvalues and relaxed conditions on the Fermi Golden rule.

Findings

Established a new theory for resonances with weaker assumptions.

Proved the uniqueness of exponential decay rates in certain cases.

Connected resonance phenomena to meromorphic continuation of Green's functions.

Abstract

Resonances which result from perturbation of embedded eigenvalues are studied by time dependent methods. A general theory is developed, with new and weaker conditions, allowing for perturbations of threshold eigenvalues and relaxed Fermi Golden rule. The exponential decay rate of resonances is addressed; its uniqueness in the time dependent picture is shown is certain cases. The relation to the existence of meromorphic continuation of the properly weighted Green's function to time dependent resonance is further elucidated, by giving an equivalent time dependent asymptotic expansion of the solutions of the Schr\"odinger equation. \keywords{Resonances; Time-dependent Schr\"odinger equation}