Special potentials for relativistic Laplacians I: Fractional Rollnik-class

TL;DR

This paper introduces a fractional Rollnik-class of potentials for relativistic Laplacians, characterizes their properties, and analyzes the spectral and self-adjointness features of associated operators, extending classical potential theory.

Contribution

It defines a fractional Rollnik-class for relativistic Laplacians, explores its properties, and studies spectral and self-adjointness aspects of related Schrödinger operators.

Findings

Coulomb-type potentials are in fractional Rollnik-class up to Hardy potential singularity.

No fractional Rollnik potential exists for $ ext{α} = 1$, but limiting cases are analyzed in low dimensions.

Extended fractional Rollnik-class is identified as the maximal space for Hilbert-Schmidt properties.

Abstract

We propose a counterpart of the classical Rollnik-class of potentials for fractional and massive relativistic Laplacians, and describe this space in terms of appropriate Riesz potentials. These definitions rely on precise resolvent estimates. We show that Coulomb-type potentials are elements of fractional Rollnik-class up to but not including the critical singularity of the Hardy potential. For the operators with fractional exponent $\alpha = 1$ there exists no fractional Rollnik potential, however, in low dimensions we make sense of these classes as limiting cases by using $\Gamma$-convergence. In a second part of the paper we derive detailed results on the self-adjointness and spectral properties of relativistic Schr\"odinger operators obtained under perturbations by fractional Rollnik potentials. We also define an extended fractional Rollnik-class which is the maximal space for the Hilbert-Schmidt property of the related Birman-Schwinger operators.