Embedded Eigenvalues and Neumann-Wigner Potentials for Relativistic Schrodinger Operators

TL;DR

This paper constructs specific potentials for relativistic Schrödinger operators that admit embedded eigenvalues, extending classical results to the relativistic case and exploring their limits and decay behaviors.

Contribution

It introduces Neumann-Wigner type potentials for relativistic Schrödinger operators with embedded eigenvalues, including in the massless case, and analyzes their properties and limits.

Findings

Constructed potentials with embedded eigenvalues in relativistic operators.

Showed convergence to classical potentials in the non-relativistic limit.

Identified unique decay behaviors due to non-locality of the operator.

Abstract

The existence of potentials for relativistic Schrodinger operators allowing eigenvalues embedded 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.