Publication: On the Schrödinger Equation for Time-Dependent Hamiltonians with a Constant Form Domain
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Publication date
2022-01-11
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MDPI AG
Abstract
We study two seminal approaches, developed by B. Simon and J. Kisynski, to the wellposedness
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.
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Keywords
Schrödinger equation, Time-dependent hamiltonian, Hilbert scales, Time-dependent domain
Bibliographic citation
Balmaseda, A., Lonigro, D., & Pérez-Pardo, J. M. (2022). On the Schrödinger Equation for Time-Dependent Hamiltonians with a Constant Form Domain. In Mathematics, 10(2), 218-238