Spectral properties of the logarithmic Laplacian

We obtain spectral inequalities and asymptotic formulae for the discrete spectrum of the operator $\frac12\, \log(-Δ)$ in an open set $Ω\in\Bbb R^d$, $d\ge2$, of finite measure with Dirichlet boundary conditions. We also derive some results regarding lower bounds for the eigenvalue $λ_1(Ω)$ and compare them with previously known inequalities.

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