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arXiv:2608.05338

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

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

math-phmath.NAquant-phSubmitted Aug 5, 2026• Announced Aug 7, 2026
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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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10 pages, 3 figures

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Many systems of interest at the forefront of technological development, for example, in quantum computation, consist of weakly interacting elements that obey quantum mechanics. A challenge for modern theoretical physics is to develop a systematic methodology for approaching these “open” quantum systems. In particular, applied mathematicians have formulated the method of transparent boundary conditions (TBCs) to model interacting quantum systems of infinite extent. Here, we consider a particular application of TBCs to the escape of a particle from a one-dimensional infinite well when one of the bounding barriers is extinguished. We analytically obtain the exact time-dependent wave function by a Green's function method and then show that a numerical solution based on the TBC method is in excellent agreement. This work was motivated by an experiment carried out at the University of Texas on escape from a gravitational wedge billiard. Physicists have used billiards to understand and explore both classical and quantum chaos. Although the original experiment was carried out in the classical regime, current work is probing lower temperatures, where a quantum mechanical formulation is required.
In a seminal work, Horsley and collaborators [S. A. R. Horsley, et al., Nat. Photon. 9, 436 (2015)] showed that, in the framework of non-Hermitian extensions of the Schr\"odinger and Helmholtz equations, a localized complex scattering potential with spatial distributions of the real and imaginary parts related to one another by the spatial Kramers-Kronig relations are reflectionless and even invisible under certain conditions. Here we consider the scattering properties of Kramers-Kronig potentials for the discrete version of the Schr\"odinger equation, which generally describes wave transport on a lattice. Contrary to the continuous Schr\"odinger equation, on a lattice a stationary Kramers-Kronig potential is reflective. However, it is shown that a slow drift can make the potential invisible under certain conditions.
Families of discrete quantum models that describe a free non-relativistic quantum particle propagating on rescaled and shifted dual weight lattices inside closures of Weyl alcoves are developed. The boundary conditions of the presented discrete quantum billiards are enforced by precisely positioned Dirichlet and Neumann walls on the borders of the Weyl alcoves. The amplitudes of the particle’s propagation to neighbouring positions are determined by a complex-valued dual-weight hopping function of finite support. The discrete dual-weight Hamiltonians are obtained as the sum of specifically constructed dual-weight hopping operators. By utilising the generalised dual-weight Fourier–Weyl transforms, the solutions of the time-independent Schrödinger equation together with the eigenenergies of the quantum systems are exactly resolved. The matrix Hamiltonians, stationary states and eigenenergies of the discrete models are exemplified for the rank two cases C2 and G2.