The Logarithmic Schr\"odinger operator and Associated Dirichlet Problems

In this note, we study the integrodifferential operator $(I-\Delta)^{\log}$ corresponding to the logarithmic symbol $\log(1+|\xi|^2)$, which is a singular integral operator given by $$ (I-\Delta)^{\log} u(x)=d_{N}\int_{\mathbb{R}^N}\frac{u(x)-u(x+y)}{|y|^{N}}\omega(|y|)\, dy, $$ where $d_N=\pi^{-\frac{N}{2}}$, $\omega(r)=2^{1-\frac{N}{2}}r^{\frac{N}{2}}K_{\frac{N}{2}}(r)$ and $K_{\nu}$ is the modified Bessel function of second kind with index $\nu$. This operator is the L\'evy generator of the variance gamma process and arises as derivative $\partial_s\Big|_{s=0}(I-\Delta)^s$ of fractional relativistic Schr\"{o}dinger operators at $s=0$. In order to study associated Dirichlet problems in bounded domains, we first introduce the functional analytic framework and some properties related to $(I-\Delta)^{\log}$, which allow to characterize the induced eigenvalue problem and Faber-Krahn type inequality. We also derive a decay estimate in $\mathbb{R}^N$ of the Poisson problem and investigate small order asymptotics $s\to 0^+$ of Dirichlet eigenvalues and eigenfunctions of $(I-\Delta)^s$ in a bounded open Lipschitz set.