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arXiv:2607.00306

Scattering, bound states, and resonances in the one-dimensional Dirac equation via supersymmetric quantum mechanics

Camila C. Soares, Luis B. Castro, Antonio S. de Castro

quant-phhep-thmath-phSubmitted Jul 1, 2026• Announced Jul 2, 2026
arXivHTMLPDF

Abstract

We develop a unified treatment of scattering and discrete spectra for the one-dimensional Dirac equation with scalar and vector interactions. Under the spin-symmetry condition, the coupled first-order Dirac system maps exactly onto an effective Sturm--Liouville (Schrö\-din\-ger-like) problem for a single spinor component. This mapping provides a convenient framework for analyzing transmission, reflection, and analytic continuation. As an explicit application, we consider effective interactions of hyperbolic Pöschl--Teller type and exploit supersymmetric quantum mechanics and shape invariance to obtain a closed-form expression for the transmission probability. The bound-state spectrum is then recovered from the poles of the analytically continued transmission amplitude, reproducing known results and offering a unified description of scattering and bound states. For the barrier configuration, we briefly comment on the resulting pole pattern in the complex momentum plane and its connection with resonance and quasi-normal-mode behavior. Moreover, we use the chiral transformation to relate the spin- and pseudospin-symmetry sectors and translate results between them without repeating the full derivation.

Journal reference
Eur. Phys. J. C 86 (2026) 1028
DOI
10.1140/epjc/s10052-026-16228-4
Comments
13 pages, 3 figures

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The Dirac equation in one dimension with a Lorentz scalar potential is associated with a supersymmetric pair of Schr\"odinger Hamiltonians H1{\mathit{H}}_{1}H1​ and H2{\mathit{H}}_{2}H2​. The H1{\mathit{H}}_{1}H1​ and H2{\mathit{H}}_{2}H2​ share the same energy spectrum and scattering phases. The shared spectrum includes the lowest states unless the Dirac equation allows a zero mode (a zero-energy bound state). This situation is unlike the common examples of supersymmetric quantum mechanics. The Dirac equation admits a zero mode only if the scalar potential has certain ``topology.'' Various such features are illustrated through explicit examples. In particular, the phase equivalence between H1{\mathit{H}}_{1}H1​ and H2{\mathit{H}}_{2}H2​ is exploited to construct transparent potentials for the Dirac equation in one dimension.
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