Heun-function analysis of the Dirac spinor spectrum in a sine-Gordon soliton background

TL;DR

This paper analyzes the Dirac spinor spectrum in a sine-Gordon soliton background using Heun functions, providing a unified approach to bound and scattering states with explicit dependence on soliton parameters.

Contribution

It introduces a systematic Heun-function based method to study Dirac spectra in soliton backgrounds, unifying bound and scattering state analysis.

Findings

Explicit spectral data expressed via Wronskians and Heun solutions

Unified treatment of bound and scattering states

Analytic and numerical verification of wave functions

Abstract

We study the Dirac spectrum in a sine-Gordon soliton background, where the induced position-dependent mass reduces the spectral problem to a Heun-type differential equation. Bound and scattering sectors are treated within a unified framework, with spectral data encoded in Wronskians matching local Heun solutions and exhibiting explicit dependence on the soliton parameters and the bare fermion mass. This formulation enables a systematic analysis of spinor bound and scattering states, supported by analytic and numerical verification of wave function matching across the soliton domain. The present work is related to arXiv:2512.07658 and emphasizes a pedagogical treatment of scattering states within the Heun-equation formalism.