Resonances under Rank One Perturbations

TL;DR

This paper investigates the generation and instability of resonances caused by rank one perturbations of selfadjoint operators, demonstrating decay properties and applications to Sturm-Liouville problems using advanced spectral analysis tools.

Contribution

It introduces new insights into resonance behavior under rank one perturbations and applies these results to Sturm-Liouville operators, expanding understanding of spectral stability.

Findings

Eigenvalues embedded in the continuous spectrum are unstable under rank one perturbations.

Resonant states exhibit almost exponential decay.

Results are applicable to Sturm-Liouville operators, linking spectral theory to differential operators.

Abstract

We study resonances generated by rank one perturbations of selfadjoint operators with eigenvalues embedded in the continuous spectrum. Instability of these eigenvalues is analyzed and almost exponential decay for the associated resonant states is exhibited. We show how these results can be applied to Sturm-Liouville operators. Main tools are the Aronszajn-Donoghue theory for rank one perturbations, a reduction process of the resolvent based on Feshbach-Livsic formula, the Fermi golden rule and a careful analysis of the Fourier transform of quasi-Lorentzian functions. We relate these results to sojourn time estimates and spectral concentration phenomena