Uniform resolvent estimates and absence of eigenvalues for Lam\'e operators with potentials

TL;DR

This paper proves that certain small, variationally controlled potentials do not introduce eigenvalues into the spectrum of the Lamé operator, and establishes uniform resolvent estimates, extending known spectral properties of the free operator.

Contribution

It demonstrates the absence of eigenvalues for perturbed Lamé operators with small potentials, including complex-valued ones, and provides uniform resolvent estimates using a multipliers technique.

Findings

No point spectrum for small perturbations of Lamé operators.

Uniform resolvent estimates are established for the perturbed operators.

Results extend spectral properties to complex-valued potentials.

Abstract

We consider the $0$-order perturbed Lam\'e operator $-\Delta^\ast + V(x)$. It is well known that if one considers the free case, namely $V=0,$ the spectrum of $-\Delta^\ast$ is purely continuous and coincides with the non-negative semi-axis. The first purpose of the paper is to show that, at least in part, this spectral property is preserved in the perturbed setting. Precisely, developing a suitable multipliers technique, we will prove the absence of point spectrum for Lam\'e operator with potentials which satisfy a variational inequality with suitable small constant. We stress that our result also covers complex-valued perturbation terms. Moreover the techniques used to prove the absence of eigenvalues enable us to provide uniform resolvent estimates for the perturbed operator under the same assumptions about $V$.