Point-like perturbed fractional Laplacians through shrinking potentials of finite range

TL;DR

This paper studies how fractional Laplacians are affected by point-like perturbations modeled as shrinking potentials, analyzing different regimes and defining zero-energy resonance for fractional Schrödinger operators.

Contribution

It provides a rigorous reconstruction of singular point-like perturbations of fractional Laplacians via shrinking regular potentials, including analysis of resonance regimes and zero-energy resonance.

Findings

Reconstruction of point-like perturbations in fractional Laplacians.

Analysis of resonance-driven and resonance-independent limits.

Qualification of zero-energy resonance for fractional Schrödinger operators.

Abstract

We reconstruct the rank-one, singular (point-like) perturbations of the $d$-dimensional fractional Laplacian in the physically meaningful norm-resolvent limit of fractional Schr\"{o}dinger operators with regular potentials centred around the perturbation point and shrinking to a delta-like shape. We analyse both the possible regimes, the resonance-driven and the resonance-independent limit, depending on the power of the fractional Laplacian and the spatial dimension. To this aim, we also qualify the notion of zero-energy resonance for Schr\"{o}dinger operators formed by a fractional Laplacian and a regular potential.