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Dynamical formulation of low-energy scattering in one dimension

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Loran, Farhang

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The transfer matrix 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 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 Schrodinger 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 Schrodinger 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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American Institute of Physics (AIP) Publishing

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Physics

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Journal of Mathematical Physics

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10.1063/5.0050990

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