Volume 24, Issue 4
Computing Eigenvalues and Eigenfunctions of Schrödinger Equations Using a Model Reduction Approach

Shuangping Li & Zhiwen Zhang

Commun. Comput. Phys., 24 (2018), pp. 1073-1100.

Published online: 2018-06

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  • Abstract

We present a model reduction approach to construct problem dependent basis functions and compute eigenvalues and eigenfunctions of stationary Schrödinger equations. The basis functions are defined on coarse meshes and obtained through solving an optimization problem. We shall show that the basis functions span a lowdimensional generalized finite element space that accurately preserves the lowermost eigenvalues and eigenfunctions of the stationary Schrödinger equations. Therefore, our method avoids the application of eigenvalue solver on fine-scale discretization and offers considerable savings in solving eigenvalues and eigenfunctions of Schr ödinger equations. The construction of the basis functions are independent of each other; thus our method is perfectly parallel. We also provide error estimates for the eigenvalues obtained by our new method. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method, especially Schrödinger equations with double well potentials are tested.

  • Keywords

Schrödinger equation, eigenvalue problems, model reduction, two-level techniques, problem dependent basis functions, computational chemistry.

  • AMS Subject Headings

35J10, 65F15, 65N25, 65N30

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COPYRIGHT: © Global Science Press

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@Article{CiCP-24-1073, author = {}, title = {Computing Eigenvalues and Eigenfunctions of Schrödinger Equations Using a Model Reduction Approach}, journal = {Communications in Computational Physics}, year = {2018}, volume = {24}, number = {4}, pages = {1073--1100}, abstract = {

We present a model reduction approach to construct problem dependent basis functions and compute eigenvalues and eigenfunctions of stationary Schrödinger equations. The basis functions are defined on coarse meshes and obtained through solving an optimization problem. We shall show that the basis functions span a lowdimensional generalized finite element space that accurately preserves the lowermost eigenvalues and eigenfunctions of the stationary Schrödinger equations. Therefore, our method avoids the application of eigenvalue solver on fine-scale discretization and offers considerable savings in solving eigenvalues and eigenfunctions of Schr ödinger equations. The construction of the basis functions are independent of each other; thus our method is perfectly parallel. We also provide error estimates for the eigenvalues obtained by our new method. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method, especially Schrödinger equations with double well potentials are tested.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.2018.hh80.08}, url = {http://global-sci.org/intro/article_detail/cicp/12319.html} }
TY - JOUR T1 - Computing Eigenvalues and Eigenfunctions of Schrödinger Equations Using a Model Reduction Approach JO - Communications in Computational Physics VL - 4 SP - 1073 EP - 1100 PY - 2018 DA - 2018/06 SN - 24 DO - http://dor.org/10.4208/cicp.2018.hh80.08 UR - https://global-sci.org/intro/cicp/12319.html KW - Schrödinger equation, eigenvalue problems, model reduction, two-level techniques, problem dependent basis functions, computational chemistry. AB -

We present a model reduction approach to construct problem dependent basis functions and compute eigenvalues and eigenfunctions of stationary Schrödinger equations. The basis functions are defined on coarse meshes and obtained through solving an optimization problem. We shall show that the basis functions span a lowdimensional generalized finite element space that accurately preserves the lowermost eigenvalues and eigenfunctions of the stationary Schrödinger equations. Therefore, our method avoids the application of eigenvalue solver on fine-scale discretization and offers considerable savings in solving eigenvalues and eigenfunctions of Schr ödinger equations. The construction of the basis functions are independent of each other; thus our method is perfectly parallel. We also provide error estimates for the eigenvalues obtained by our new method. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method, especially Schrödinger equations with double well potentials are tested.

Shuangping Li & Zhiwen Zhang. (2020). Computing Eigenvalues and Eigenfunctions of Schrödinger Equations Using a Model Reduction Approach. Communications in Computational Physics. 24 (4). 1073-1100. doi:10.4208/cicp.2018.hh80.08
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