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Nonperturbative time-dependent model of scattering a particle on a one-dimensional $δ$-potential

N. L. Chuprikov

TL;DR

This work presents a nonperturbative, time-dependent analysis of a particle scattering from a one-dimensional $\delta$-potential, showing that wave operators are inapplicable and that the scattering space splits into superselection sectors by both momentum sign and localization relative to the barrier. The study reveals two classes of states: pure interactions lasting indefinitely and mixed scattering states whose asymptotics are governed by a combined asymptotic superselection rule, causing decoherence in a closed system. Explicit stationary and nonstationary solutions are derived, and it is shown that certain asymptotes can be free-like yet describe interaction with the barrier, while a restricted $k$-domain (e.g., $k<0$ absence) yields conventional transmission and reflection channels within invariant sectors. The results argue for a revised scattering framework for 1D short-range potentials, where scattering must be treated as decohering, channel-based processes (transmission and reflection) rather than a universal pure scattering state connected to free dynamics.

Abstract

It is shown that the concept of wave operators is inapplicable to the description of the quantum dynamics of a particle in the field of a one-dimensional $δ$-potential (1D-$δ$P). The solutions of the Schrödinger equation in this problem are divided into two classes: some are pure states that describe the process of interaction of a particle with the 1D-$δ$P, which lasts infinitely long; others describe the scattering process, but are mixed states, in the space of which there acts, at $t\to\mp\infty$, a combined asymptotic rule of superselection with respect to the sign of the particle momentum and the region of localization of its state (to the left or to the right of the 1D-$δ$P); the Hamiltonian with the 1D-$δ$P is defined only in superselection sectors; the scattering process with one-sided incidence of a particle on the 1D-$δ$P is the process of decoherence in a closed system.

Nonperturbative time-dependent model of scattering a particle on a one-dimensional $δ$-potential

TL;DR

This work presents a nonperturbative, time-dependent analysis of a particle scattering from a one-dimensional -potential, showing that wave operators are inapplicable and that the scattering space splits into superselection sectors by both momentum sign and localization relative to the barrier. The study reveals two classes of states: pure interactions lasting indefinitely and mixed scattering states whose asymptotics are governed by a combined asymptotic superselection rule, causing decoherence in a closed system. Explicit stationary and nonstationary solutions are derived, and it is shown that certain asymptotes can be free-like yet describe interaction with the barrier, while a restricted -domain (e.g., absence) yields conventional transmission and reflection channels within invariant sectors. The results argue for a revised scattering framework for 1D short-range potentials, where scattering must be treated as decohering, channel-based processes (transmission and reflection) rather than a universal pure scattering state connected to free dynamics.

Abstract

It is shown that the concept of wave operators is inapplicable to the description of the quantum dynamics of a particle in the field of a one-dimensional -potential (1D-P). The solutions of the Schrödinger equation in this problem are divided into two classes: some are pure states that describe the process of interaction of a particle with the 1D-P, which lasts infinitely long; others describe the scattering process, but are mixed states, in the space of which there acts, at , a combined asymptotic rule of superselection with respect to the sign of the particle momentum and the region of localization of its state (to the left or to the right of the 1D-P); the Hamiltonian with the 1D-P is defined only in superselection sectors; the scattering process with one-sided incidence of a particle on the 1D-P is the process of decoherence in a closed system.
Paper Structure (9 sections, 59 equations)

This paper contains 9 sections, 59 equations.

Table of Contents

  1. Introduction
  2. Stationary and non-stationary states describing the quantum dynamics of a particle in the field of a one-dimensional $\delta$-potential
  3. The general solution $\Psi_L$ as a pure quantum state describing the process of interaction of a particle with a $\delta$-potential
  4. On the link between the norm of the general solution $\Psi_L$ and the function $\mathcal{A}_L(k)$
  5. Asymptotes of the general solution $\Psi_L$
  6. Asymptotes of the general solution $\Psi_L$ when the function $\mathcal{A}_L(k)$ is identically equal to zero at $k<0$
  7. On the absence of transitions between the subspaces $\widehat{\mathcal{S}}_{(+)}$ and $\widehat{\mathcal{S}}_{(-)}$ under the action of observables' operators
  8. Combined asymptotic superselection rule
  9. Conclusion