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Volume 38, Issue 3
Absorbing Boundary Conditions for Variable Potential Schrödinger Equations via Titchmarsh-Weyl Theory

Matthias Ehrhardt & Chunxiong Zheng

DOI: 10.4208/cicp.OA-2024-0178

Commun. Comput. Phys., 38 (2025), pp. 711-728.

Published online: 2025-08

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  • Abstract

We propose a novel approach to simulate the solution of the time-dependent Schrödinger equation with a general variable potential. The key idea is to approximate the Titchmarsh-Weyl $m$-function (exact Dirichlet-to-Neumann operator) by a rational function with respect to an appropriate spectral parameter. By using this method, we overcome the usual high-frequency restriction associated with absorbing boundary conditions in general variable potential problems. The resulting fast computational algorithm for absorbing boundary conditions ensures accuracy over the entire frequency range.

  • Keywords

Absorbing boundary conditions, variable potential, Schrödinger equation, Titchmarsh-Weyl $m$-function, unbounded domain.

  • AMS Subject Headings

65M99, 81-08

  • Copyright

COPYRIGHT: © Global Science Press

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  • TXT
@Article{CiCP-38-711, author = {Ehrhardt , Matthias and Zheng , Chunxiong}, title = {Absorbing Boundary Conditions for Variable Potential Schrödinger Equations via Titchmarsh-Weyl Theory}, journal = {Communications in Computational Physics}, year = {2025}, volume = {38}, number = {3}, pages = {711--728}, abstract = {

We propose a novel approach to simulate the solution of the time-dependent Schrödinger equation with a general variable potential. The key idea is to approximate the Titchmarsh-Weyl $m$-function (exact Dirichlet-to-Neumann operator) by a rational function with respect to an appropriate spectral parameter. By using this method, we overcome the usual high-frequency restriction associated with absorbing boundary conditions in general variable potential problems. The resulting fast computational algorithm for absorbing boundary conditions ensures accuracy over the entire frequency range.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2024-0178}, url = {http://global-sci.org/intro/article_detail/cicp/24313.html} }
TY - JOUR T1 - Absorbing Boundary Conditions for Variable Potential Schrödinger Equations via Titchmarsh-Weyl Theory AU - Ehrhardt , Matthias AU - Zheng , Chunxiong JO - Communications in Computational Physics VL - 3 SP - 711 EP - 728 PY - 2025 DA - 2025/08 SN - 38 DO - http://doi.org/10.4208/cicp.OA-2024-0178 UR - https://global-sci.org/intro/article_detail/cicp/24313.html KW - Absorbing boundary conditions, variable potential, Schrödinger equation, Titchmarsh-Weyl $m$-function, unbounded domain. AB -

We propose a novel approach to simulate the solution of the time-dependent Schrödinger equation with a general variable potential. The key idea is to approximate the Titchmarsh-Weyl $m$-function (exact Dirichlet-to-Neumann operator) by a rational function with respect to an appropriate spectral parameter. By using this method, we overcome the usual high-frequency restriction associated with absorbing boundary conditions in general variable potential problems. The resulting fast computational algorithm for absorbing boundary conditions ensures accuracy over the entire frequency range.

Ehrhardt , Matthias and Zheng , Chunxiong. (2025). Absorbing Boundary Conditions for Variable Potential Schrödinger Equations via Titchmarsh-Weyl Theory. Communications in Computational Physics. 38 (3). 711-728. doi:10.4208/cicp.OA-2024-0178
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