Absence of eigenvalues in the continuous spectrum for Klein-Gordon operators

TL;DR

This paper constructs a one-dimensional von-Neumann Wigner potential for Klein-Gordon operators, providing explicit examples of embedded eigenvalues and proving their absence in broad potential classes, including Coulomb potential.

Contribution

It introduces a novel potential construction for Klein-Gordon operators and establishes conditions for the absence of eigenvalues in the continuous spectrum.

Findings

Explicit example of Klein-Gordon operator with embedded positive eigenvalues.

Proof of absence of eigenvalues in the continuous spectrum for broad potential classes.

Expression of Klein-Gordon operator in Schrödinger form to analyze spectral regions.

Abstract

We construct the one-dimensional analogous of von-Neumann Wigner potential to the relativistic Klein-Gordon operator, in which is defined taking asymptotic mathematical rules in order to obtain existence conditions of eigenvalues embedded in the continuous spectrum. Using our constructed potential, we provide an explicit and analytical example of the Klein-Gordon operator with positive eigenvalues embedded in the so called relativistic "continuum region". Even so in this not standard example, we present the region of the "continuum" where those eigenvalues cannot occur. Besides, the absence of eigenvalues in the continuous spectrum for Klein-Gordon operators is proven to a broad general potential classes, including the minimally coupled electric Coulomb potential. Considering known techniques available in literature for Schrodinger operators, we demonstrate an expression for Klein-Gordon operator written in Schrodinger's form, whereby is determined the mathematical spectrum region of absence of eigenvalues.