The logarithmic Schr\"odinger operator and associated Dirichlet problems

In this note, we study the integrodifferential operator $(I-\mathrm{\Delta }{)}^{\log }$ corresponding to the logarithmic symbol $\log (1+\mathrm{\mid }\xi {\mathrm{\mid }}^2)$, which is a singular integral operator given by $$(I-\mathrm{\Delta }{)}^{\log }u(x)=d_N{\int }_{{\mathbb{R}}^N}\frac{u(x)-u(x+y)}{\mathrm{\mid }y{\mathrm{\mid }}^N}\omega (\mathrm{\mid }y\mathrm{\mid })dy,$$ 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 ${\mathrm{\partial }}_s{\mid }_{s=0}(I-\mathrm{\Delta }{)}^s$ 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 $(I-\mathrm{\Delta }{)}^{\log }$, 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 $(I-\mathrm{\Delta }{)}^s$ in a bounded open Lipschitz set.