On a scattering length for additive functionals and spectrum of fractional Laplacians with non-local perturbations

TL;DR

This paper investigates the scattering length for additive functionals of symmetric stable processes and links it to the spectral properties of fractional Laplacians with non-local perturbations, extending previous results to non-continuous functionals.

Contribution

It extends the concept of scattering length to non-continuous additive functionals and establishes a criterion for the discreteness of the spectrum of fractional Laplacians with non-local measure perturbations.

Findings

Semi-classical limit of scattering length equals the capacity of the measure's support.

Provides an equivalent criterion for discrete spectrum based on scattering length.

Connects scattering length with the bottom of the spectrum of Schrödinger operators.

Abstract

In this paper we study the scattering length for positive additive functionals of symmetric stable processes on ${\bf R}^d$. The additive functionals considered here are not necessarily continuous. We prove that the semi-classical limit of the scattering length equals the capacity of the support of a certain measure potential, thus extend previous results for the case of positive continuous additive functionals. We also give an equivalent criterion for the fractional Laplacian with a measure valued non-local operator as a perturbation to have purely discrete spectrum in terms of the scattering length, by considering the connection between scattering length and the bottom of the spectrum of Schr\"odinger operator in our settings.