On the Schrödinger Equation for Time-Dependent Hamiltonians with a Constant Form Domain
We study two seminal approaches, developed by B. Simon and J. Kisyński, to the well-posedness of the Schrödinger equation with a time-dependent Hamiltonian. In both cases, the Hamiltonian is assumed to be semibounded from below and to have a constant form domain, but a possibly non-constant operator...
| Κύριοι συγγραφείς: | , , |
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| Μορφή: | Άρθρο |
| Γλώσσα: | English |
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MDPI AG
2022-01-01
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| Σειρά: | Mathematics |
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| Διαθέσιμο Online: | https://www.mdpi.com/2227-7390/10/2/218 |
| _version_ | 1827664454970507264 |
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| author | Aitor Balmaseda Davide Lonigro Juan Manuel Pérez-Pardo |
| author_facet | Aitor Balmaseda Davide Lonigro Juan Manuel Pérez-Pardo |
| author_sort | Aitor Balmaseda |
| collection | DOAJ |
| description | We study two seminal approaches, developed by B. Simon and J. Kisyński, to the well-posedness of the Schrödinger equation with a time-dependent Hamiltonian. In both cases, the Hamiltonian is assumed to be semibounded from below and to have a constant form domain, but a possibly non-constant operator domain. The problem is addressed in the abstract setting, without assuming any specific functional expression for the Hamiltonian. The connection between the two approaches is the relation between sesquilinear forms and the bounded linear operators representing them. We provide a characterisation of the continuity and differentiability properties of form-valued and operator-valued functions, which enables an extensive comparison between the two approaches and their technical assumptions. |
| first_indexed | 2024-03-10T01:01:16Z |
| format | Article |
| id | doaj.art-190f0f79dd4e47b091748c26b51b08a1 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T01:01:16Z |
| publishDate | 2022-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-190f0f79dd4e47b091748c26b51b08a12023-11-23T14:34:09ZengMDPI AGMathematics2227-73902022-01-0110221810.3390/math10020218On the Schrödinger Equation for Time-Dependent Hamiltonians with a Constant Form DomainAitor Balmaseda0Davide Lonigro1Juan Manuel Pérez-Pardo2Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Madrid, SpainDipartimento di Fisica and MECENAS, Università di Bari, I-70126 Bari, ItalyDepartamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Madrid, SpainWe study two seminal approaches, developed by B. Simon and J. Kisyński, to the well-posedness of the Schrödinger equation with a time-dependent Hamiltonian. In both cases, the Hamiltonian is assumed to be semibounded from below and to have a constant form domain, but a possibly non-constant operator domain. The problem is addressed in the abstract setting, without assuming any specific functional expression for the Hamiltonian. The connection between the two approaches is the relation between sesquilinear forms and the bounded linear operators representing them. We provide a characterisation of the continuity and differentiability properties of form-valued and operator-valued functions, which enables an extensive comparison between the two approaches and their technical assumptions.https://www.mdpi.com/2227-7390/10/2/218Schrödinger equationtime-dependent HamiltonianHilbert scalestime-dependent domain |
| spellingShingle | Aitor Balmaseda Davide Lonigro Juan Manuel Pérez-Pardo On the Schrödinger Equation for Time-Dependent Hamiltonians with a Constant Form Domain Mathematics Schrödinger equation time-dependent Hamiltonian Hilbert scales time-dependent domain |
| title | On the Schrödinger Equation for Time-Dependent Hamiltonians with a Constant Form Domain |
| title_full | On the Schrödinger Equation for Time-Dependent Hamiltonians with a Constant Form Domain |
| title_fullStr | On the Schrödinger Equation for Time-Dependent Hamiltonians with a Constant Form Domain |
| title_full_unstemmed | On the Schrödinger Equation for Time-Dependent Hamiltonians with a Constant Form Domain |
| title_short | On the Schrödinger Equation for Time-Dependent Hamiltonians with a Constant Form Domain |
| title_sort | on the schrodinger equation for time dependent hamiltonians with a constant form domain |
| topic | Schrödinger equation time-dependent Hamiltonian Hilbert scales time-dependent domain |
| url | https://www.mdpi.com/2227-7390/10/2/218 |
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