Transparent boundary conditions for the spatially discrete Schrödinger equation: Reflectionless quantum transport in 1D lattices

• URL: http://arxiv.org/abs/2608.05338v1
• Date: Wed, 05 Aug 2026 18:54:46 GMT
• Title: Transparent boundary conditions for the spatially discrete Schrödinger equation: Reflectionless quantum transport in 1D lattices
• Abstract summary: We construct exact transparent boundary conditions (TBCs) for a time-continuous, spatially discrete Schrdinger equation.<n>We rigorously demonstrate the consistency of this discrete formulation with its continuous counterpart in the continuum limit.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We construct exact transparent boundary conditions (TBCs) for a time-continuous, spatially discrete Schrödinger equation that models a one-dimensional quantum lattice. Using a recently developed exact solution for the discrete system, we derive the Dirichlet-to-Neumann maps analytically via Laplace transforms. This yields a convolution-type boundary condition governed by Bessel functions. We rigorously demonstrate the consistency of this discrete formulation with its continuous counterpart in the continuum limit. Additionally, we present an efficient time-discretization scheme based on the trapezoidal rule for practical implementation. Numerical experiments using a Crank-Nicolson solver verify that our proposed TBCs eliminate spurious backscattering entirely and preserve reflectionless propagation of a Gaussian wave packet exiting the computational domain

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