Spectral properties of the logarithmic Laplacian

Abstract

We obtain spectral inequalities and asymptotic formulae for the discrete spectrum of the operator $\frac{1}{2}\log (-\Delta )$ in an open set $\Omega \in {\mathbb{R}}^d$, $d\ge 2$, 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.

21 pages, 1 figure