Fall-off of eigenfunctions for non-local Schrödinger operators with decaying potentials

2016-06-07T10:20:36Z (GMT) by Kamil Kaleta Jozsef Lorinczi
We study the spatial decay of eigenfunctions of non-local Schrodinger operators based on generators of symmetric jump-paring Levy 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 Levy 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 Levy intensity. We furthermore prove that under reasonable conditions the Levy intensity also governs the upper bounds of eigenfunctions, and a ground state is comparable to it by two-sided bounds. As an interesting consequence, we identify a sharp regime change in the decay of eigenfunctions as the Levy intensity is varied from sub-exponential to exponential order, and dependent on the location of the eigenvalue, in the sense that through the transition Levy intensity-driven decay becomes slower than the rate of Levy intensity. Our approach is based on path integration and probabilistic potential theory techniques, and all results are also illustrated by specific examples.