Simplicity of extremal eigenvalues of the Klein-Gordon equation

TL;DR

This paper investigates the extremal eigenvalues of the Klein-Gordon equation with unbounded electric potentials, proving their simplicity and positivity under specific spectral conditions, with illustrative examples provided.

Contribution

It establishes the simplicity and positivity of extremal eigenvalues for a class of Klein-Gordon spectral problems, expanding understanding of their spectral properties.

Findings

Extremal eigenvalues are simple and have positive eigenfunctions.

Spectral conditions involve the spectrum being in two disjoint intervals with boundary points as eigenvalues.

Examples of potentials satisfying these spectral conditions are provided.

Abstract

We consider the spectral problem associated with the Klein-Gordon equation for unbounded electric potentials. If the spectrum of this problem is contained in two disjoint real intervals and the two inner boundary points are eigenvalues, we show that these extremal eigenvalues are simple and possess strictly positive eigenfunctions. Examples of electric potentials satisfying these assumptions are given.