Spectral Properties of the Logarithmic Laplacian with Indefinite Weights

TL;DR

This paper studies the spectral properties of the logarithmic Laplacian with indefinite weights, establishing eigenvalue existence, simplicity, sign properties, inequalities, and monotonicity.

Contribution

It introduces new spectral results for the logarithmic Laplacian with indefinite weights, including eigenvalue characterization and domain and weight monotonicity.

Findings

Existence of an unbounded sequence of eigenvalues.

First eigenvalue is simple with a constant sign eigenfunction.

Higher eigenfunctions change sign and satisfy a nodal domain inequality.

Abstract

In this paper, we investigate a weighted eigenvalue problem driven by the Logarithmic Laplacian with indefinite weights. We prove the existence of an unbounded sequence of Lusternik-Schnirelman eigenvalues and show that the first eigenvalue is simple, with the associated eigenfunction having constant sign in the domain. In contrast, eigenfunctions corresponding to higher eigenvalues necessarily change sign. We further establish a nodal domain type inequality relating the higher eigenvalues to the measure of the positive and negative parts of the corresponding eigenfunctions, which is of independent interest. As an application, we prove that the first eigenvalue is isolated. In addition, we obtain alternative variational characterizations of the first and second eigenvalues and establish monotonicity properties of the eigenvalues with respect to both the weight function and the domain.