Resonance near a doubly degenerate embedded eigenvalue

TL;DR

This paper investigates resonance phenomena near doubly degenerate embedded eigenvalues of the Laplacian, introducing a new topological approach to analyze spectral density and related scattering properties.

Contribution

It introduces a novel application of the Morse Lemma to study rank-two perturbations and extends resonance analysis to the case of double degeneracy.

Findings

Asymptotic spectral density near degenerate eigenvalues

Behavior of scattering cross-section and time delay

Handling of threshold eigenvalues

Abstract

This paper extends the study of resonance phenomenon initiated by the authors in~\cite{LS} to the case of doubly degenerate embedded eigenvalues (i.e. eigenvalue of multiplicity two). A fundamentally new concept is introduced to resolve the difficulties that arise in this study, beyond the methods of \cite{LS}. We apply a differential topological technique, namely the Morse Lemma, to study the present case. This allows us to understand rank-two self-adjoint perturbations of the Laplacian on $L^{2}(\mathbb{R}^{3})$, and along with methods of \cite{LS}, we obtain asymptotic results for the spectral density near a doubly degenerate embedded eigenvalue. Importantly, we are able to easily handle the threshold eigenvalue case. \par We also analyze important properties which explain such resonance phenomenon, viz., asymptotic behaviour of the sojourn time, scattering cross-section and time delay.