Improvements to the computation of eigenvalues and eigenfunctions of two-dimensional Schrödinger equations by constant perturbation based algorithms
We present important improvements and additions to a modern technique developed by Ixaru to solve the time-dependent two-dimensional Schrödinger equation with homogeneous Dirichlet boundary conditions over a rectangular domain. The algorithm, first described in Ixaru (2010), is based on the so-calle...
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| Published in: | Journal of computational and applied mathematics Vol. 412; p. 114292 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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Elsevier B.V
01.10.2022
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| ISSN: | 0377-0427, 1879-1778 |
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| Abstract | We present important improvements and additions to a modern technique developed by Ixaru to solve the time-dependent two-dimensional Schrödinger equation with homogeneous Dirichlet boundary conditions over a rectangular domain. The algorithm, first described in Ixaru (2010), is based on the so-called Constant Perturbation technique. In this paper, we refine and extend the algorithm with important features. We focus in particular on new algorithms for the determination of the index of the eigenvalues, for the orthonormalization of eigenfunctions, for automatic step size selection and for the accurate computation of integrals. We provide the new developments with sufficient theoretical background and numerical experiments. |
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| AbstractList | We present important improvements and additions to a modern technique developed by Ixaru to solve the time-dependent two-dimensional Schrödinger equation with homogeneous Dirichlet boundary conditions over a rectangular domain. The algorithm, first described in Ixaru (2010), is based on the so-called Constant Perturbation technique. In this paper, we refine and extend the algorithm with important features. We focus in particular on new algorithms for the determination of the index of the eigenvalues, for the orthonormalization of eigenfunctions, for automatic step size selection and for the accurate computation of integrals. We provide the new developments with sufficient theoretical background and numerical experiments. |
| ArticleNumber | 114292 |
| Author | Van Daele, Marnix Baeyens, Toon |
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| Cites_doi | 10.1145/2839299 10.1016/j.cpc.2006.09.004 10.1007/BF01386087 10.1007/BF01206624 10.1006/jcph.1996.0140 10.1016/S0010-4655(98)00181-7 10.1145/1114268.1114273 10.1007/BF02142741 10.1016/j.cpc.2012.12.016 10.1016/j.cpc.2010.06.031 10.1016/j.apnum.2004.08.018 10.1016/S0010-4655(02)00459-9 |
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| References | Ixaru (b1) 2010; 181 Braun, Sofianos, Papageorgiou, Lagaris (b19) 1996; 126 Ixaru (b2) 1984 Marletta (b18) 1993; 4 Ixaru (b6) 2002; 147 Ixaru, De Meyer, Vanden Berghe (b3) 1999; 118 Van Daele, Vanden Berghe, Vande Vyver (b11) 2005; 53 Atkinson (b17) 1964 Courant, Hilbert (b15) 2008 Baeyens, Van Daele (b9) 2020 Titchmarsh (b10) 1962 Ledoux, Van Daele, Vanden Berghe (b4) 2005; 31 Piessens, de Doncker-Kapenga, Überhuber, Kahaner (b12) 1983 Ledoux, Van Daele (b7) 2013; 184 Lancaster (b13) 1964; 6 Ledoux, Van Daele, Vanden Berghe (b8) 2007; 176 Prüfer (b14) 1926; 95 Hale (b16) 2005 Ledoux, Van Daele, Vanden Berghe (b5) 2016; 42 Van Daele (10.1016/j.cam.2022.114292_b11) 2005; 53 Lancaster (10.1016/j.cam.2022.114292_b13) 1964; 6 Marletta (10.1016/j.cam.2022.114292_b18) 1993; 4 Braun (10.1016/j.cam.2022.114292_b19) 1996; 126 Ixaru (10.1016/j.cam.2022.114292_b6) 2002; 147 Titchmarsh (10.1016/j.cam.2022.114292_b10) 1962 Piessens (10.1016/j.cam.2022.114292_b12) 1983 Atkinson (10.1016/j.cam.2022.114292_b17) 1964 Hale (10.1016/j.cam.2022.114292_b16) 2005 Ixaru (10.1016/j.cam.2022.114292_b2) 1984 Ixaru (10.1016/j.cam.2022.114292_b1) 2010; 181 Ledoux (10.1016/j.cam.2022.114292_b4) 2005; 31 Prüfer (10.1016/j.cam.2022.114292_b14) 1926; 95 Ledoux (10.1016/j.cam.2022.114292_b8) 2007; 176 Ledoux (10.1016/j.cam.2022.114292_b7) 2013; 184 Courant (10.1016/j.cam.2022.114292_b15) 2008 Ixaru (10.1016/j.cam.2022.114292_b3) 1999; 118 Ledoux (10.1016/j.cam.2022.114292_b5) 2016; 42 Baeyens (10.1016/j.cam.2022.114292_b9) 2020 |
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| SubjectTerms | Constant Perturbations Methods Eigenfunctions Eigenvalues Schrödinger equation |
| Title | Improvements to the computation of eigenvalues and eigenfunctions of two-dimensional Schrödinger equations by constant perturbation based algorithms |
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