Embedded Eigenvalues and NeumannWigner Potentials for Relativistic Schrodinger Operators
Abstract
The existence of potentials for relativistic Schrodinger operators allowing eigenvalues embedded in the essential spectrum is a longstanding open problem. We construct NeumannWigner type potentials for the massive relativistic Schrodinger operator in one and three dimensions for which an embedded eigenvalue exists. We show that in the nonrelativistic limit these potentials converge to the classical NeumannWigner and MosesTuan 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 zeroresonance exists, dependent on the rate of decay of the corresponding eigenfunctions. We obtain explicit formulae and observe unusual decay behaviours due to the nonlocality of the operator.
 Publication:

arXiv eprints
 Pub Date:
 April 2016
 arXiv:
 arXiv:1605.00196
 Bibcode:
 2016arXiv160500196L
 Keywords:

 Mathematical Physics;
 47A75;
 47G30 (Primary);
 34L40;
 47A40;
 81Q10 (Secondary)
 EPrint:
 J.Funct.Anal., vol. 273, Issue 4 (2017)