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Physica Scripta

Analysis of discontinuous wave packets interacting with one dimensional singular potentials

Victor Barrera-Figueroa, Manuel Gadella, Miguel Angel Díaz-Cigales

Published May 8, 2026• 0 citations

Abstract

Abstract We investigate the dynamical effects induced by contact interactions on the propagation of quantum wave packets within the framework of the one-dimensional Schrödinger equation. The analysis is carried out using the spectral parameter power series (SPPS) method, which provides a unified approach to treat both regular and singular potentials. Particular attention is devoted to the role of contact interactions in shaping scattering dynamics. We show that such interactions produce distinctive features, including time-dependent discontinuities in the wave function, anisotropic reflection depending on the direction of incidence, and a non-trivial high- energy behavior in which the transmission coefficient does not converge to unity. The method is validated against exactly solvable models, including the finite potential barrier and a truncated version of the Pöschl-Teller potential, where excellent agreement with analytical results is observed. These results demonstrate both the robustness of the SPPS method and the qualitative differences between regular and singular interactions in quantum scattering.

DOI: 10.1088/1402-4896/ae6ad5

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Abstract We investigate the dynamical effects induced by contact interactions on the propagation of quantum wave packets within the framework of the one-dimensional Schrödinger equation. The analysis is carried out using the spectral parameter power series (SPPS) method, which provides a unified approach to treat both regular and singular potentials. Particular attention is devoted to the role of contact interactions in shaping scattering dynamics. We show that such interactions produce distinctive features, including time-dependent discontinuities in the wave function, anisotropic reflection depending on the direction of incidence, and a non-trivial high- energy behavior in which the transmission coefficient does not converge to unity. The method is validated against exactly solvable models, including the finite potential barrier and a truncated version of the Pöschl-Teller potential, where excellent agreement with analytical results is observed. These results demonstrate both the robustness of the SPPS method and the qualitative differences between regular and singular interactions in quantum scattering.

One-dimensional quantum scattering from a local potential barrier is considered. Analytical properties of the scattering amplitudes have been investigated by means of the integral equations equivalent to the Schrödinger equations. The transition and reflection amplitudes are expressed in terms of two complex functions of the incident energy, which are similar to the Jost function in partial-wave scattering. These functions are entire for finite-range potentials and meromorphic for exponentially decreasing potentials. The analytical properties result from the locality of the potential in the wave equation and represent the effect of causality in the time dependence of the scattering process.

One-dimensional quantum scattering from a local potential barrier is considered. Analytical properties of the scattering amplitudes have been investigated by means of the integral equations equivalent to the Schrodinger equations. The transition and reflection amplitudes are expressed in terms of two complex functions of the incident energy, which are similar to the Jost function in the partial-wave scattering. These functions are entire for finite-range potentials and meromorphic for exponentially decreasing potentials. The analytical properties result from locality of the potential in the wave equation and represent the effect of causality in time dependence of the scattering process.

We discuss a solution of the time-dependent Schrödinger equation that incorporates absorbing boundary conditions and a method for extracting the reflection and transmission probabilities for wave packets interacting with time-dependent potential barriers. We apply the method to a rectangular barrier that moves with constant velocity, an oscillating rectangular barrier, a locally periodic barrier with an amplitude modulated by a traveling wave, and a locally periodic potential with an amplitude modulated by a standing wave. Visualizations of the reflection phenomena are presented with an emphasis on understanding these systems from their dynamics. Applications to non-stationary neutron optics experiments are discussed briefly.

The spectral singularity (SS) from a non-Hermitian potential is one of the most remarkable scattering feature of non-Hermitian quantum mechanics. At the spectral singular point, the scattering amplitudes diverge to infinite. This phenomena have been extensively studied over the last two decades. The previous studies have suggested the need of extremely fine control of the various system parameters for practical applications of SS. However no study have been carried out to understand the behavior of the scattering amplitude in the vicinity of SS. Such study would be important towards the practical application of SS where it is desired to maintain outgoing scattering amplitude to a specific value. The behavior of the loci of constant scattering amplitude in the neighborhood of SS are studied for a pure imaginary barrier potential

iV

,

V \in R^{+}

. We show that these loci are elliptical and self-similar to each other in

E-V

plane where

E

is energy of the wave. The orientations of these ellipses for reflection (

R

) and transmission (

T

) amplitude around a given SS are same. This is surprising due to the different mathematical forms of

R

and

T

. In the vicinity of SS,

R \approx T

.

The scattered wave formalism developed for a quantum subsystem interacting with reservoirs through open boundaries is applied to one- or two-dimensional barrier scattering and quantum transistors. The total wave function is divided into incident and scattered components. Markovian outgoing wave boundary conditions are imposed on the scattered or total wave function by either the ratio or polynomial methods. For barrier scattering problems, accurate time-dependent transmission probabilities are obtained through the integration of the modified time-dependent Schrödinger equations for the scattered wave function. For quantum transistors, the time-dependent transport is studied for a quantum wave packet propagating through the conduction channel of a field effect transistor. This study shows that the scattered wave formalism significantly reduces computational effort relative to other open boundary methods and demonstrates wide applications to quantum dynamical processes.