Dynamical formulation of low-energy scattering in one dimension

• URL: http://arxiv.org/abs/2102.06084v1
• Date: Thu, 11 Feb 2021 15:55:34 GMT
• Title: Dynamical formulation of low-energy scattering in one dimension
• Authors: Farhang Loran and Ali Mostafazadeh
• Abstract summary: A transfer matrix $mathbfM$ of a short-range potential may be expressed in terms of the time-evolution operator for an effective two-level quantum system. We explore the utility of this formulation in the study of the low-energy behavior of the scattering data.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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.

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