Spectral properties of the logarithmic Laplacian

Abstract We obtain spectral inequalities and asymptotic formulae for the discrete spectrum of operator $$\frac{1}{2}\, \log (-\Delta )$$ 1 2 log ( - ? ) in an open set $$\Omega \in \mathbb R^d$$ ? ? R d , $$d\ge 2$$ ? finite measure with Dirichlet boundary conditions. also derive some results regarding lower bounds eigenvalue $$\lambda _1(\Omega ? compare them previously known inequalities.