On fractional powers of singular perturbations of the Laplacian

TL;DR

This paper characterizes fractional powers of the Hamiltonian with point interactions in three dimensions, detailing their domains, decompositions, and associated Sobolev space norms.

Contribution

It provides explicit descriptions of the domains and decompositions of fractional singular perturbations of the Laplacian with point interactions.

Findings

Explicit domain descriptions for fractional Hamiltonians

Decomposition into regular and singular parts

Control of singular fractional Sobolev norms

Abstract

We qualify a relevant range of fractional powers of the so-called Hamiltonian of point interaction in three dimensions, namely the singular perturbation of the negative Laplacian with a contact interaction supported at the origin. In particular we provide an explicit control of the domain of such a fractional operator and of its decomposition into regular and singular parts. We also qualify the norms of the resulting singular fractional Sobolev spaces and their mutual control with the corresponding classical Sobolev norms.