Embedded eigenvalues and Neumann–Wigner potentials for relativistic Schrödinger operators
journal contributionposted on 2017-04-11, 13:06 authored by Jozsef Lorinczi, Itaru Sasaki
The existence of potentials for relativistic Schrodinger operators allowing eigenvalues em bedded in the essential spectrum is a long-standing open problem. We construct Neumann-Wigner type potentials for the massive relativistic Schrodinger operator in one and three dimensions for which an embedded eigenvalue exists. We show that in the non-relativistic limit these potentials converge to the classical Neumann-Wigner and Moses-Tuan potentials, respectively. For the massless operator in one dimension we construct two families of potentials, different by the parities of the (generalized) eigenfunctions, for which an eigenvalue equal to zero or a zero-resonance exists, dependent on the rate of decay of the corresponding eigenfunctions. We obtain explicit formulae and observe unusual decay behaviours due to the non-locality of the operator.
- Mathematical Sciences
Published inJournal of Functional Analysis
Pages1548 - 1575
CitationLORINCZI, J. and SASAKI, I., 2017. Embedded eigenvalues and Neumann-Wigner potentials for relativistic Schroedinger operators. Journal of Functional Analysis, 273 (4), pp. 1548-1575.
- AM (Accepted Manuscript)
Publisher statementThis 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/
NotesThis paper was accepted for publication in the journal Journal of Functional Analysis and the definitive published version is available at http://dx.doi.org/10.1016/j.jfa.2017.03.012