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Fall-off of eigenfunctions for non-local Schrödinger operators with decaying potentials
journal contribution
posted on 2016-06-07, 10:20 authored by Kamil Kaleta, Jozsef LorincziWe 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.
Funding
K. Kaleta was supported by the National Science Center (Poland) grant 2012/04/S/ST1/00093 and by the Foundation for Polish Science.
History
School
- Science
Department
- Mathematical Sciences
Published in
Potential AnalysisVolume
46Issue
4Pages
647 - 688Citation
KALETA, K. and LORINCZI, J., 2016. Fall-off of eigenfunctions for non-local Schrödinger operators with decaying potentials. Potential Analysis, 46 (4), pp. 647–688.Publisher
© Springer Science+Business Media DordrechtVersion
- AM (Accepted Manuscript)
Publisher statement
This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/Acceptance date
2016-09-20Publication date
2016-10-25Notes
The final publication is available at link.springer.com via http://dx.doi.org/10.1007/s11118-016-9597-3. An ArXiv pre-print is available online at: http://arxiv.org/abs/1503.03508v2ISSN
0926-2601eISSN
1572-929XPublisher version
Other identifier
1503.03508Language
- en