Dynamical formulation of low-energy scattering in one dimension

TL;DR

This paper presents a dynamical approach to low-energy scattering in one dimension, using transfer matrices and effective two-level quantum systems, enabling systematic series expansions and resonance identification.

Contribution

It introduces a novel dynamical formulation of stationary scattering and develops iterative schemes for low-energy series expansions using transfer matrices.

Findings

Series expansion of transfer matrix in powers of wavenumber

Perturbative solutions for zero-energy Schrödinger equation

Method to identify zero-energy resonances via transfer matrix zeros

Abstract

The transfer matrix ${\mathbf{M}}$ of a short-range potential may be expressed in terms of the time-evolution operator for an effective two-level quantum system with a time-dependent non-Hermitian Hamiltonian. This leads to a dynamical formulation of stationary scattering. We explore the utility of this formulation in the study of the low-energy behavior of the scattering data. In particular, for the exponentially decaying potentials, we devise a simple iterative scheme for computing terms of arbitrary order in the series expansion of ${\mathbf{M}}$ in powers of the wavenumber. The coefficients of this series are determined in terms of a pair of solutions of the zero-energy stationary Schr\"odinger equation. We introduce a transfer matrix for the latter equation, express it in terms of the time-evolution operator for an effective two-level quantum system, and use it to obtain a perturbative series expansion for the solutions of the zero-energy stationary Schr\"odinger equation. Our approach allows for identifying the zero-energy resonances for scattering potentials in both full line and half-line with zeros of the entries of the zero-energy transfer matrix of the potential or its trivial extension to the full line.