Special comparison theorem for the Dirac equation

TL;DR

This paper presents a special comparison theorem for the Dirac equation, showing that if the potential is monotone in a parameter, then the eigenvalues are also monotone, extending previous results to all discrete eigenvalues.

Contribution

It introduces a new comparison theorem for the Dirac equation that applies to all discrete eigenvalues under monotonic potential conditions, generalizing earlier node-free state results.

Findings

Eigenvalues are monotone with respect to the potential parameter a.

The theorem applies to every discrete eigenvalue, not just node-free states.

Generalizes previous comparison theorems for the Dirac equation.

Abstract

If a central vector potential V(r,a) in the Dirac equation is monotone in a parameter 'a', then a discrete eigenvalue E(a) is monotone in 'a'. For such a special class of comparisons, this generalizes an earlier comparison theorem that was restricted to node free states. Moreover, the present theorem applies to every discrete eigenvalue.