Fractional powers and singular perturbations of quantum differential Hamiltonians

TL;DR

This paper investigates the mathematical properties of fractional powers and singular perturbations of quantum Hamiltonians, focusing on their resolvent structures and domain singularities, with implications for Schrödinger equations.

Contribution

It introduces a comparative analysis of two constructions of singular perturbations of fractional Laplacians, highlighting their perturbative and local singularity features.

Findings

Comparison of resolvent structures for the two constructions

Analysis of local singularity structures of domains

Outline of future research directions in Schrödinger equations

Abstract

We consider the fractional powers of singular (point-like) perturbations of the Laplacian, and the singular perturbations of fractional powers of the Laplacian, and we compare such two constructions focusing on their perturbative structure for resolvents and on the local singularity structure of their domains. In application to the linear and non-linear Schr\"{o}dinger equations for the corresponding operators we outline a programme of relevant questions that deserve being investigated.