Resolvent bounds for jump generators and ground state asymptotics for nonlocal Schr\"{o}dinger operators

TL;DR

This paper establishes bounds on the resolvent of jump generator operators with convolution kernels and analyzes the asymptotic behavior of ground states and solutions to related nonlocal Schrödinger equations.

Contribution

It provides new resolvent bounds for jump generators with exponential or polynomial decay kernels and applies these to eigenfunction estimates and evolution equations.

Findings

Resolved bounds for jump generator resolvents.

Derived pointwise eigenfunction estimates.

Analyzed solution behavior for nonlocal Schrödinger equations.

Abstract

The paper deals with jump generators with a convolution kernel. Assuming that the kernel decays either exponentially or polynomially we prove a number of lower and upper bounds for the resolvent of such operators. We consider two applications of these results. First we obtain pointwise estimates for principal eigenfunction of jump generators perturbed by a compactly supported potential (so-called nonlocal Schr\"odinger operators). Then we consider the Cauchy problem for the corresponding inhomogeneous evolution equations and study the behaviour of its solutions.