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

TL;DR

This work addresses the Dirac spinor spectrum in a sine-Gordon kink background, mapping the coupled Dirac equations to Heun-type differential equations to treat both bound and scattering sectors within a unified framework. A Wronskian matching procedure yields the scattering coefficients and encodes spectral data in the Heun parameters, while analytic continuation of the scattering solutions reveals the bound-state spectrum via zeros of $c_1$. The spinor phase shifts are found to be identical for the upper and lower components and obey Levinson's theorem, linking bound states to the scattering phase. The approach provides a systematic, exact framework with explicit dependence on the soliton parameters and bare mass $M$, offering a template for Heun-based spectral analyses in other solitonic backgrounds and related field theories.

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.

Figures (7)

• Figure 1: (color online) Scalar kink (blue) for $M=1.5, \beta =1.$
• Figure 2: (color online) Scalar kink (blue) in the left and right Figs. The real and imaginary parts of the incident, reflected and transmitted scattering wave function $u(x)$: Left Fig. $Re[c_1 u_{in}(x)]$ (green), $Re[c_2 u_{ref}(x)]$ (brown), $Re[u_{tr}(x)]$ (red), and right Fig. $Im[c_1 u_{in}(x)]$ (green) and $Im[c_2 u_{ref}(x)]$ (brown), $Im[u_{tr}(x)]$ (red). For $\beta =1, M=5, k=2.5, E_1=+5.6$.
• Figure 3: (color online) Scalar kink (blue) around $x=0$. The real part of the scattering wave function $u(x)$ for $x<0$, $Re[c_1 u_{in}(x) +c_2 u_{ref}(x)]$ (green), and for $x > 0$, $Re[u_{tr}(x)]$ (red). The inset figure shows the smooth matching at $x=0$.
• Figure 4: (color online) Scalar kink (blue) around $x=0$. The imaginary part of the scattering wave function $u(x)$ for $x<0$, $Re[c_1 u_{in}(x) +c_2 u_{ref}(x)]$ (green), and for $x > 0$, $Im[u_{tr}(x)]$ (red). The inset figure shows the smooth matching at $x=0$.
• Figure 5: (color online) Scalar kink (blue) around $x=0$. The real part of the scattering wave function $v(x)$ for $x<0$, $Re[c_1 v_{in}(x) +c_2 v_{ref}(x)]$ (green), and for $x > 0$, $Re[v_{tr}(x)]$ (red). The inset figure shows the smooth matching at $x=0$.