Decay of resolvent kernels and Schr\"odinger eigenstates for L\'evy operators

TL;DR

This paper analyzes how the resolvent kernels and Schr"odinger eigenstates decay spatially for a broad class of non-local L"evy operators, revealing sharp transitions based on the decay properties of the L"evy measures.

Contribution

It provides new sharp decay estimates for resolvent kernels and bound states of L"evy operators, extending previous results to more general non-local operators.

Findings

Identifies sharp decay transitions for resolvent kernels based on L"evy measure decay.

Extends decay analysis from fractional Laplacians to broader L"evy operators.

Uses operator semigroup methods to derive decay estimates.

Abstract

We study the spatial decay behaviour of resolvent kernels for a large class of non-local L\'evy operators and bound states of the corresponding Schr\"odinger operators. Our findings naturally lead us to proving results for L\'evy measures, which have subexponential or exponential decay, respectively. This leads to sharp transitions in the the decay rates of the resolvent kernels. We obtain estimates that allow us to describe and understand the intricate decay behaviour of the resolvent kernels and the bound states in either regime, extending findings by Carmona, Masters and Simon for fractional Laplacians (the subexponential regime) and classical relativistic operators (the exponential regime). Our proofs are mainly based on methods from the theory of operator semigroups.