Bethe-Salpeter Equations with Instantaneous Confinement: Establishing Stability of Bound States

TL;DR

This paper investigates the conditions under which Bethe-Salpeter equations with confining potentials produce stable, real, and discrete bound state spectra, emphasizing the importance of the Lorentz structure of the kernels.

Contribution

It systematically identifies confining Bethe-Salpeter kernels that ensure stable bound states with real, discrete spectra, addressing instabilities in numerical solutions.

Findings

Analytic criteria for stable bound states with confining interactions.

Identification of Lorentz structures that prevent instabilities.

Conditions for real and discrete energy spectra in Bethe-Salpeter equations.

Abstract

Salpeter equations with potential functions rising to infinity in configuration space do not automatically predict stable bound states. For this to happen, also the Lorentz behaviour of the involved Bethe-Salpeter kernels is crucial. At least for interaction potentials of harmonic-oscillator form analytic scrutinies of Salpeter equations with confining interactions may identify those Bethe-Salpeter kernels which describe bound states free from the notorious instabilities encountered in numerical evaluations. Such truly confining kernels can be singled out systematically by requesting the resulting bound-state energy spectra to be both real and discrete.