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

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.

13 pages, 3 figures

References in corpus (16)

• Supersymmetry and Quantum Mechanics
• Quasinormal modes of black holes and black branes
• Pseudo-spin as a Relativistic Symmetry
• Quasi-normal modes of Schwarzschild-de Sitter black holes
• Quasinormal modes of d-dimensional spherical black holes with a near extreme cosmological constant
• The supersymmetric modified Poschl-Teller and delta-well potentials
• Role of black hole quasinormal mode overtones for ringdown analysis
• Resonances and antibound states of Pöschl-Teller potential: Ladder operators and SUSY partners
• Relativistic Effects of Mixed Vector-Scalar-Pseudoscalar Potentials for Fermions in 1+1 Dimensions
• Bounded solutions of fermions in the background of mixed vector-scalar Pöschl-Teller-like potentials
• Highly-damped quasi-normal frequencies for piecewise Eckart potentials
• General spin and pseudospin symmetries of the Dirac equation
• Pseudospin and spin symmetries in 1+1 dimensions: The case of the Coulomb potential
• Reflectionless PT-symmetric potentials in the one-dimensional Dirac equation
• Noncommutative quasinormal modes of Schwarzschild black hole
• Quasinormal modes of Plebański-Demiański black hole in the near-Nariai regime