Uniform resolvent estimates and absence of eigenvalues of biharmonic operators with complex potentials

TL;DR

This paper establishes conditions under which the spectrum of biharmonic operators with complex potentials remains stable, providing explicit resolvent estimates and extending results to critical potentials in higher dimensions.

Contribution

It introduces explicit stability conditions for the spectrum of biharmonic operators with complex perturbations, including critical potentials, and derives uniform resolvent estimates in weighted spaces.

Findings

Spectrum stability under certain complex perturbations

Explicit resolvent estimates in weighted spaces

Extension of results to critical Rellich-type potentials

Abstract

We quantify the subcriticality of the bilaplacian in dimensions greater than four by providing explicit repulsivity/smallness conditions on complex additive perturbations under which the spectrum remains stable. Our assumptions cover critical Rellich-type potentials too. As a byproduct we obtain uniform resolvent estimates in weighted spaces. Some of the results are new also in the self-adjoint setting.