Dynamical resonances and SSF singularities for a magnetic Schroedinger operator

TL;DR

This paper studies how embedded eigenvalues of a 3D magnetic Schrödinger operator turn into resonances under perturbations, analyzing their asymptotic behavior, time evolution, and spectral shift function singularities.

Contribution

It provides new asymptotic expansions for resonances, conditions for Fermi Golden Rule validity, and insights into the spectral shift function's singularities for perturbed magnetic Schrödinger operators.

Findings

Asymptotic expansion of resonances as perturbation tends to zero.

Identification of dense sets of perturbations satisfying Fermi Golden Rule.

Analysis of spectral shift function singularities at embedded eigenvalues.

Abstract

We consider the Hamiltonian $H$ of a 3D spinless non-relativistic quantum particle subject to parallel constant magnetic and non-constant electric field. The operator $H$ has infinitely many eigenvalues of infinite multiplicity embedded in its continuous spectrum. We perturb $H$ by appropriate scalar potentials $V$ and investigate the transformation of these embedded eigenvalues into resonances. First, we assume that the electric potentials are dilation-analytic with respect to the variable along the magnetic field, and obtain an asymptotic expansion of the resonances as the coupling constant $\varkappa$ of the perturbation tends to zero. Further, under the assumption that the Fermi Golden Rule holds true, we deduce estimates for the time evolution of the resonance states with and without analyticity assumptions; in the second case we obtain these results as a corollary of suitable Mourre estimates and a recent article of Cattaneo, Graf and Hunziker \cite{cgh}. Next, we describe sets of perturbations $V$ for which the Fermi Golden Rule is valid at each embedded eigenvalue of $H$; these sets turn out to be dense in various suitable topologies. Finally, we assume that $V$ decays fast enough at infinity and is of definite sign, introduce the Krein spectral shift function for the operator pair $(H+V, H)$, and study its singularities at the energies which coincide with eigenvalues of infinite multiplicity of the unperturbed operator $H$.