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arXiv · 2608.05338

Transparent boundary conditions for the spatially discrete Schr\"odinger equation: Reflectionless quantum transport in 1D lattices

Mashrab E. Akramov·Jambul R. Yusupov·Matthias Ehrhardt·Davron U. Matrasulov
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Abstract

We construct exact transparent boundary conditions (TBCs) for a time-continuous, spatially discrete Schr\"odinger 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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Mashrab E. Akramov, Jambul R. Yusupov, Matthias Ehrhardt, Davron U. Matrasulov. 2026-08-05. Transparent boundary conditions for the spatially discrete Schr\"odinger equation: Reflectionless quantum transport in 1D lattices. https://arxiv.org/abs/2608.05338

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