Eigenvalue bounds and spectral stability of Lam\'e operators with complex potentials

TL;DR

This paper establishes bounds on eigenvalues of non self-adjoint Lamé operators with complex potentials, providing conditions for spectral stability and showing the spectrum remains purely continuous under certain perturbations.

Contribution

It offers new quantitative bounds on eigenvalues and demonstrates spectral stability for Lamé operators with complex potentials in three dimensions.

Findings

Eigenvalue bounds depend on potential norms

Spectrum remains purely continuous under specific perturbations

No embedded eigenvalues under certain conditions

Abstract

This paper is devoted to providing quantitative bounds on the location of eigenvalues, both discrete and embedded, of non self-adjoint Lam\'e operators of elasticity $-\Delta^\ast + V$ in terms of suitable norms of the potential $V$. In particular, this allows to get sufficient conditions on the size of the potential such that the point spectrum of the perturbed operator remains empty. In three dimensions we show full spectral stability under suitable form-subordinated perturbations: we prove that the spectrum is purely continuous and coincides with the non negative semi-axis as in the free case.