aiqbench analysis
This research advances the mathematical framework of quantum simulation by deriving exact transparent boundary conditions for discrete Schrödinger equations, effectively eliminating spurious backscattering in 1D lattice models. By rigorously proving consistency between discrete formulations and their continuum limits using Laplace transforms and Bessel functions, the study provides a robust foundation for more accurate numerical implementations of quantum transport phenomena. While highly relevant to computational physics, this work represents fundamental theoretical progress rather than an immediate product or policy shift in applied AI/quantum sectors.
Transparent boundary conditions for the spatially discrete Schr\"odinger equation: Reflectionless quantum transport in 1D lattices
arXiv:2608.05338v1 Announce Type: cross 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
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
Mathematical Physics (math-ph); Numerical Analysis (math.NA); Quantum Physics (quant-ph)
arXiv:2608.05338 [math-ph]
(or arXiv:2608.05338v1 [math-ph] for this version)
https://doi.org/10.48550/arXiv.2608.05338 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 209 words
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