# Bounds on eigenvalues of perturbed Lam\'e operators with complex potentials

## TL;DR

This paper improves bounds on the eigenvalues of perturbed Lamé operators with complex potentials, advancing understanding of their spectral properties and extending results to non self-adjoint cases.

## Contribution

It provides new quantitative bounds on the discrete spectrum of Lamé operators with complex potentials, including extensions to self-adjoint cases.

## Key findings

- Improved bounds on eigenvalues for non self-adjoint Lamé operators
- Quantitative spectral bounds in terms of $L^p$-norms of potentials
- Extension of results to self-adjoint Lamé operators

## Abstract

Several recent papers have focused their attention in proving the correct analogue to the Lieb-Thirring inequalities for non self-adjoint operators and in finding bounds on the distribution of their eigenvalues in the complex plane. This paper provides some improvement in the state of the art in this topic. Precisely, we address the question of finding quantitative bounds on the discrete spectrum of the perturbed Lam\'e operator of elasticity $-\Delta^\ast + V$ in terms of $L^p$-norms of the potential. Original results within the self-adjoint framework are provided too.