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, dge 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.