Volume 24, Issue 4
A Hybrid Method for Computing the Schrödinger Equations with Periodic Potential with Band-Crossings in the Momentum Space

Lihui Chai, Shi Jin & Peter A. Markowich

Commun. Comput. Phys., 24 (2018), pp. 989-1020.

Published online: 2018-06

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  • Abstract

We propose a hybrid method which combines the Bloch decompositionbased time splitting (BDTS) method and the Gaussian beam method to simulate the Schrödinger equation with periodic potentials in the case of band-crossings. With the help of the Bloch transformation, we develop a Bloch decomposition-based Gaussian beam (BDGB) approximation in the momentum space to solve the Schrödinger equation. Around the band-crossing a BDTS method is used to capture the inter-band transitions, and away from the crossing, a BDGB method is applied in order to improve the efficiency. Numerical results show that this method can capture the inter-band transitions accurately with a computational cost much lower than the direct solver. We also compare the Schrödinger equation with its Dirac approximation, and numerically show that, as the rescaled Planck number ε→0, the Schrödinger equation converges to the Dirac equations when the external potential is zero or small, but for general external potentials there is an O(1) difference in between the solutions of the Schrödinger equation and its Dirac approximation.

  • Keywords

Schrödinger equation, band-crossing, Dirac point, Bloch decomposition, time-splitting spectral method, Gaussian beam method.

  • AMS Subject Headings

35Q41, 65M70, 81Q20, 74Q10, 35B27

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-24-989, author = {}, title = {A Hybrid Method for Computing the Schrödinger Equations with Periodic Potential with Band-Crossings in the Momentum Space}, journal = {Communications in Computational Physics}, year = {2018}, volume = {24}, number = {4}, pages = {989--1020}, abstract = {

We propose a hybrid method which combines the Bloch decompositionbased time splitting (BDTS) method and the Gaussian beam method to simulate the Schrödinger equation with periodic potentials in the case of band-crossings. With the help of the Bloch transformation, we develop a Bloch decomposition-based Gaussian beam (BDGB) approximation in the momentum space to solve the Schrödinger equation. Around the band-crossing a BDTS method is used to capture the inter-band transitions, and away from the crossing, a BDGB method is applied in order to improve the efficiency. Numerical results show that this method can capture the inter-band transitions accurately with a computational cost much lower than the direct solver. We also compare the Schrödinger equation with its Dirac approximation, and numerically show that, as the rescaled Planck number ε→0, the Schrödinger equation converges to the Dirac equations when the external potential is zero or small, but for general external potentials there is an O(1) difference in between the solutions of the Schrödinger equation and its Dirac approximation.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.2018.hh80.01}, url = {http://global-sci.org/intro/article_detail/cicp/12315.html} }
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