The logarithmic Schrödinger operator and associated Dirichlet problems

Abstract

In this note, we study the integrodifferential operator (I−Δ)log corresponding to the logarithmic symbol log⁡(1+|ξ|2), which is a singular integral operator given by(I−Δ)logu(x)=dN∫RNu(x)−u(x+y)|y|Nω(|y|)dy, where dN=π−N2, ω(r)=21−N2rN2KN2(r) and Kν is the modified Bessel function of second kind with index ν. This operator is the Lévy generator of the variance gamma process and arises as derivative ∂s|s=0(I−Δ)s of fractional relativistic Schrödinger 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−Δ)log, which allow to characterize the induced eigenvalue problem and Faber-Krahn type inequality. We also derive a decay estimate in RN of the Poisson problem and investigate small order asymptotics s→0+ of Dirichlet eigenvalues and eigenfunctions of (I−Δ)s in a bounded open Lipschitz set.