Stability in the instantaneous Bethe-Salpeter formalism: harmonic-oscillator reduced Salpeter equation

TL;DR

This paper rigorously proves the stability of bound-state solutions in a simplified harmonic-oscillator model of the reduced Salpeter equation, addressing previous numerical instabilities linked to the Klein paradox.

Contribution

It provides an analytic spectral analysis demonstrating the stability of solutions in the harmonic-oscillator reduced Salpeter equation, a step beyond prior numerical studies.

Findings

Bound-state solutions have real, discrete energy spectra.

Solutions are bounded from below, indicating stability.

Analytic proof addresses previous numerical instabilities.

Abstract

A popular three-dimensional reduction of the Bethe-Salpeter formalism for the description of bound states in quantum field theory is the Salpeter equation, derived by assuming both instantaneous interactions and free propagation of all bound-state constituents. Numerical (variational) studies of the Salpeter equation with confining interaction, however, observed specific instabilities of the solutions, likely related to the Klein paradox and rendering (part of the) bound states unstable. An analytic investigation of this problem by a comprehensive spectral analysis is feasible for the reduced Salpeter equation with only harmonic-oscillator confining interactions. There we are able to prove rigorously that the bound-state solutions correspond to real discrete energy spectra bounded from below and are thus free of any instabilities.