The fractional logarithmic Schr\"{o}dinger operator: properties and functional spaces

TL;DR

This paper investigates the fractional logarithmic Schr"{o}dinger operator, establishing its integral representation, functional spaces, and key properties for PDE analysis, including Green functions and variational inequalities.

Contribution

It introduces a new functional analytic framework for the fractional logarithmic Schr"{o}dinger operator, including integral representation and asymptotic analysis.

Findings

Derived the integral representation of the operator

Established asymptotics of the Green function

Developed variational inequalities and PDE framework

Abstract

In this note, we deal with the fractional Logarithmic Schr\"{o}dinger operator $(I+(-\Delta)^s)^{\log}$ and the corresponding energy spaces for variational study. The fractional (relativistic) Logarithmic Schr\"{o}dinger operator is the pseudo-differential operator with logarithmic Fourier symbol, $\log(1+|\xi|^{2s})$, $s>0$. We first establish the integral representation corresponding to the operator and provide an asymptotics property of the related kernel. We introduce the functional analytic theory allowing to study the operator from a PDE point of view and the associated Dirichlet problems in an open set of $\mathbb{ R}^N.$ We also establish some variational inequalities, provide the fundamental solution and the asymptotics of the corresponding Green function at zero and at infinity.