The starting point is the Dirac equation in one spatial dimension with two distinct kinds of external interactions: a Lorentz-scalar coupling and a time-like vector coupling. These enter the theory through two combinations of the potentials, denoted Sigma and Delta, which control the emergence of the so-called spin and pseudospin symmetries. These symmetries were originally discussed in relativistic mean-field models of atomic nuclei, where they suppress spin-orbit splittings and dramatically simplify the spectrum. In the one-dimensional setting studied here, the spin-symmetry limit corresponds to setting Delta to zero, and it is under precisely this condition that the coupled first-order Dirac system decouples into a single second-order Sturm-Liouville equation for one spinor component, with an algebraic formula recovering the other component.

This reduction is more than a calculational convenience. The resulting effective potential is generally energy dependent, a hallmark of relativistic dynamics, but the equation itself has exactly the structure of a nonrelativistic scattering problem. That means the entire arsenal of nonrelativistic techniques, from transmission and reflection amplitudes to the analytic continuation of scattering data into the complex momentum plane, can be imported into the relativistic framework in a controlled way. The authors also exploit the chiral transformation, a discrete symmetry of the Dirac equation that exchanges the upper and lower spinor components, swaps the roles of Sigma and Delta, and flips the sign of the fermion mass. Thanks to this transformation, every result derived in the spin-symmetry sector can be translated into the pseudospin-symmetry sector without repeating a single line of derivation.

To make the framework concrete, the team chose one of the most celebrated profiles in theoretical physics: the hyperbolic Pöschl-Teller potential, a smooth, localized well or barrier proportional to the square of the hyperbolic secant function. First introduced in 1933, this potential has served for nearly a century as a benchmark for exactly solvable quantum problems. Its special virtue in this context is that it belongs to the class of shape-invariant potentials, for which supersymmetric quantum mechanics generates an algebraic ladder of solutions. In the supersymmetric construction, a Hamiltonian is factorized into first-order operators built from a superpotential, and a partner Hamiltonian is obtained by reversing the order of those operators. When the partner potential has the same functional form as the original, with shifted parameters, the spectra and scattering data can be obtained by pure algebra.

Applying this machinery to the Dirac-matched Pöschl-Teller system, the authors derived a closed-form expression for the transmission amplitude, written entirely in terms of gamma functions. The corresponding transmission probability takes a strikingly simple form governed by the ratio of a hyperbolic sine to a trigonometric sine. The trigonometric factor encodes the depth of the effective well through a parameter lambda that is itself fixed self-consistently by the Dirac energy, because the effective potential inherits an energy dependence from the relativistic reduction. Whenever lambda approaches an integer value, the sine factor vanishes and the potential becomes perfectly transparent: the particle sails through without any reflection. For integer lambda the well is exactly reflectionless at all energies, a result the authors recover elegantly by iterating the supersymmetric recursion until the problem reduces to a free particle.

The energy dependence of the effective coupling produces genuinely relativistic effects that have no nonrelativistic analogue. Near the positive-energy threshold, where the momentum approaches zero, the transmission probability is highly sensitive to the strength of the scalar and vector potentials, and the authors show numerically that different potential depths produce markedly different low-energy transparency. As the energy increases, the hyperbolic factor in the transmission probability rapidly dominates and the system becomes essentially transparent, as expected for any short-range interaction. The non-monotonic behavior near threshold is traced directly to the trajectory of the energy-dependent lambda parameter: when it passes close to an integer, transmission is enhanced; when it stays far from integers, reflection is stronger.

Perhaps the most satisfying aspect of the approach is that bound states emerge from the same analytic structure as scattering. By continuing the transmission amplitude to imaginary momentum, the authors identify its poles with the discrete spectrum. The gamma-function representation has simple poles at negative integers, and the pole condition yields a finite tower of square-integrable states, exactly those with quantum numbers smaller than the effective coupling lambda. Combining the pole condition with the relativistic dispersion relation reproduces the known bound-state spectrum of the Dirac-Pöschl-Teller system, providing a nontrivial consistency check and demonstrating that continuum scattering and discrete bound levels are two faces of the same analytic object.

The barrier configuration reveals an even richer structure. When the sign of the potential is flipped, the effective interaction becomes repulsive and no bound states exist. Instead, the analytically continued transmission amplitude exhibits poles in the lower half of the complex momentum plane, which are interpreted as resonances or, when selected by purely outgoing boundary conditions at both spatial infinities, as quasi-normal modes. The authors find that the qualitative pole pattern is controlled by a single discriminant. Below a critical barrier strength, the poles sit on the imaginary axis and correspond to antibound or virtual states. Above it, the effective coupling acquires an imaginary part and the poles split into two symmetric families marching into the lower half-plane at fixed real parts and equally spaced imaginary parts, with higher modes damping more rapidly. This is precisely the pole architecture familiar from Pöschl-Teller models of quasinormal ringing in wave systems, including near-extremal black holes in anti-de Sitter spacetime, where the same barrier shape appears as an effective potential for perturbations.

The connection to gravitational physics is more than decorative. Quasinormal modes govern the characteristic ringdown of perturbed black holes, and the Pöschl-Teller barrier is widely used as an analytically tractable proxy for the effective potentials that control those oscillations. By showing how the same pole classification arises in a genuinely relativistic, energy-dependent setting, the new work suggests a natural starting point for a systematic quasi-normal-mode analysis of Dirac-type problems, in which the pole conditions must be solved self-consistently in the complex energy or momentum plane. The authors point to this as a promising extension, alongside the exploration of other shape-invariant interactions within the same supersymmetric framework.

What the study ultimately delivers is a lesson in unification. A single analytic object, the transmission amplitude, encodes the transmitted and reflected fluxes of relativistic particles, the discrete bound-state spectrum through its poles on the imaginary axis, and the resonant, exponentially decaying modes through its poles in the complex plane. The chiral transformation doubles the reach of every result, connecting the spin and pseudospin sectors with a stroke of the pen. In an era when much of theoretical physics relies on heavy numerical computation, there is enduring value in problems that can be solved exactly and completely, and the one-dimensional Dirac equation with Pöschl-Teller interactions, viewed through the supersymmetric lens, has now joined their ranks in full.

Subject of Research: Exact solutions of the one-dimensional Dirac equation with scalar and vector interactions using supersymmetric quantum mechanics and shape invariance

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

Article References: Soares, C. C., Castro, L. B., & de Castro, A. S. (2026). Scattering, bound states, and resonances in the one-dimensional Dirac equation via supersymmetric quantum mechanics. The European Physical Journal C, 86(9), Article 1028. https://doi.org/10.1140/epjc/s10052-026-16228-4

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16228-4

Keywords: Dirac equation, supersymmetric quantum mechanics, shape invariance, Pöschl-Teller potential, scattering theory, bound states, resonances, quasinormal modes, spin symmetry, pseudospin symmetry, chiral transformation, transmission amplitude