Schr\"odinger Models for Solutions of the Bethe-Salpeter Equation in Minkowski Space. II. Fermionic Bound-State Constituents

TL;DR

This paper explores a method to approximate solutions of the Minkowski-space Bethe-Salpeter equation for fermionic bound states using a Schr"odinger potential model derived through geometric spectral inversion, simplifying complex relativistic calculations.

Contribution

It introduces a novel approach to model fermionic bound states by applying geometric spectral inversion to Bethe-Salpeter spectral data, accommodating singular potentials.

Findings

Potential models qualitatively replicate Bethe-Salpeter features

Spectral inversion effectively derives nonrelativistic potentials

Method handles complexities of Bethe-Salpeter formalism

Abstract

In view of the obstacles encountered in any attempts to solve the Minkowski-space Bethe-Salpeter equation for bound states of two fermions, we study the possibility to model the bound-state features, at least at a qualitative level, by a Schr\"odinger description. Such a nonrelativistic potential model can be constructed by applying, to any given Bethe-Salpeter spectral data, "geometric spectral inversion" in its recently extended form, which tolerates also singular potentials. This leads to the adaptation of explicit models that provide an overview accounting for the Bethe-Salpeter formalism's complexities.