Spectral properties of the logarithmic Laplacian

TL;DR

This paper investigates the spectral characteristics of the logarithmic Laplacian operator in bounded domains, providing inequalities, asymptotic formulas, and bounds for eigenvalues, enhancing understanding of its spectral behavior.

Contribution

It introduces new spectral inequalities and asymptotic formulas for the logarithmic Laplacian's eigenvalues, along with improved lower bounds for the first eigenvalue.

Findings

Derived spectral inequalities for the logarithmic Laplacian.

Established asymptotic formulas for the discrete spectrum.

Provided new lower bounds for the first eigenvalue.

Abstract

We obtain spectral inequalities and asymptotic formulae for the discrete spectrum of the operator $\frac12\, \log(-\Delta)$ in an open set $\Omega\in\Bbb R^d$, $d\ge2$, of finite measure with Dirichlet boundary conditions. We also derive some results regarding lower bounds for the eigenvalue $\lambda_1(\Omega)$ and compare them with previously known inequalities.