A modified Fermi Golden Rule at threshold for 3D magnetic Schr\"odinger operators

TL;DR

This paper investigates how a small magnetic field affects a threshold eigenvalue in 3D Schrödinger operators, transforming it into a resonance and providing asymptotic analysis of its properties.

Contribution

It introduces a modified Fermi Golden Rule at threshold for magnetic Schrödinger operators, revealing how magnetic fields induce resonances from eigenvalues.

Findings

Threshold eigenvalues become resonances under magnetic perturbation

Asymptotic expansion of the resonance's imaginary part is derived

Differences between magnetic and electric field induced resonances are discussed

Abstract

In this paper we consider three-dimensional Schr\"odinger operators with a simple threshold eigenvalue. We show, under certain assumptions, that when a small magnetic field is introduced, this eigenvalue turns into a resonance in the time-dependent sense. We find the leading term in the asymptotic expansion of the imaginary part of the resonance and discuss the principal differences with respect to resonances induced by weak electric fields obtained previously in the literature.