Exponential propagators for the Schrödinger equation with a time-dependent potential
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2018
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| _version_ | 1866910419708280832 |
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| author | Bader, Philipp Blanes, Sergio Kopylov, Nikita |
| author_facet | Bader, Philipp Blanes, Sergio Kopylov, Nikita |
| contents | We consider the numerical integration of the Schrödinger equation with a time-dependent Hamiltonian given as the sum of the kinetic energy and a time-dependent potential. Commutator-free (CF) propagators are exponential propagators that have shown to be highly efficient for general time-dependent Hamiltonians. We propose new CF propagators that are tailored for Hamiltonians of said structure, showing a considerably improved performance. We obtain new fourth- and sixth-order CF propagators as well as a novel sixth-order propagator that incorporates a double commutator that only depends on coordinates, so this term can be considered as cost-free. The algorithms require the computation of the action of exponentials on a vector similarly to the well known exponential midpoint propagator, and this is carried out using the Lanczos method. We illustrate the performance of the new methods on several numerical examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1804_07103 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Exponential propagators for the Schrödinger equation with a time-dependent potential Bader, Philipp Blanes, Sergio Kopylov, Nikita Numerical Analysis 65L05, 65Z05 We consider the numerical integration of the Schrödinger equation with a time-dependent Hamiltonian given as the sum of the kinetic energy and a time-dependent potential. Commutator-free (CF) propagators are exponential propagators that have shown to be highly efficient for general time-dependent Hamiltonians. We propose new CF propagators that are tailored for Hamiltonians of said structure, showing a considerably improved performance. We obtain new fourth- and sixth-order CF propagators as well as a novel sixth-order propagator that incorporates a double commutator that only depends on coordinates, so this term can be considered as cost-free. The algorithms require the computation of the action of exponentials on a vector similarly to the well known exponential midpoint propagator, and this is carried out using the Lanczos method. We illustrate the performance of the new methods on several numerical examples. |
| title | Exponential propagators for the Schrödinger equation with a time-dependent potential |
| topic | Numerical Analysis 65L05, 65Z05 |
| url | https://arxiv.org/abs/1804.07103 |