Stability of Salpeter Solutions

TL;DR

This paper investigates the stability of solutions to the Salpeter equation within quantum field theory, demonstrating analytically that the harmonic-oscillator reduced Salpeter equation produces stable bound states without instabilities.

Contribution

It provides the first analytical spectral analysis showing the stability of solutions in the harmonic-oscillator reduced Salpeter equation, addressing previous numerical instability concerns.

Findings

Solutions are free of instabilities

Analytic methods confirm stable bound states

Supports the use of simplified models for stability analysis

Abstract

In the framework of instantaneous approximations to the Bethe-Salpeter formalism for the description of bound states within quantum field theories, depending on the Lorentz structure of the Bethe-Salpeter interaction kernel the solutions of the full Salpeter equation with some confining interaction may exhibit certain instabilities, which are possibly related to the Klein paradox and signal the decay of states assumed to be bound by the confining interactions. They are observed in numerical (variational) studies of the Salpeter equation. The presumably simplest scenario allowing for the fully analytic investigation of this problem is set by the reduced Salpeter equation with harmonic-oscillator interaction. In this case, Salpeter's integral equation simplifies to either an algebraic relation or a second-order homogeneous linear ordinary differential equation, immediately accessible to standard techniques. There one may hope to be able to decide unambiguously whether this setting poses a well-defined eigenvalue problem the solutions of which correspond to stable bound states associated to real energy eigenvalues bounded from below. By analytical spectral analysis the bound-state solutions of this "harmonic-oscillator reduced Salpeter equation" can be shown to be free of such instabilities.