Fall-off of eigenfunctions for non-local Schr\"odinger operators with decaying potentials

TL;DR

This paper investigates how eigenfunctions of non-local Schr"odinger operators decay at infinity, revealing that their decay rates are governed by the underlying Lévy process and potential, with sharp conditions and regime changes identified.

Contribution

It establishes sharp decay bounds for eigenfunctions of non-local Schr"odinger operators with Lévy process generators, linking decay rates to jump intensities and potential eigenvalues.

Findings

Eigenfunctions decay at most as rapidly as the Lévy intensity.

Decay becomes slower than Lévy intensity if jump-paring property fails.

Ground states are comparable with Lévy intensity, with two-sided bounds.

Abstract

We study the spatial decay of eigenfunctions of non-local Schr\"odinger operators whose kinetic terms are generators of symmetric jump-paring L\'evy processes with Kato-class potentials decaying at infinity. This class of processes has the property that the intensity of single large jumps dominates the intensity of all multiple large jumps. We find that the decay rates of eigenfunctions depend on the process via specific preference rates in particular jump scenarios, and depend on the potential through the distance of the corresponding eigenvalue from the edge of the continuous spectrum. We prove that the conditions of the jump-paring class imply that for all eigenvalues the corresponding positive eigenfunctions decay at most as rapidly as the L\'evy intensity. This condition is sharp in the sense that if the jump-paring property fails to hold, then eigenfunction decay becomes slower than the decay of the L\'evy intensity. We furthermore prove that under reasonable conditions the L\'evy intensity also governs the upper bounds of eigenfunctions, and ground states are comparable with it, i.e., two-sided bounds hold. As an interesting consequence, we identify a sharp regime change in the decay of eigenfunctions as the L\'evy intensity is varied from sub-exponential to exponential order, and dependent on the location of the eigenvalue, in the sense that through the transition L\'evy intensity-driven decay becomes slower than the rate of decay of L\'evy intensity. Our approach is based on path integration and probabilistic potential theory techniques, and all results are also illustrated by specific examples.