On the Schrödinger Equation for Time-DependentHamiltonians with a Constant Form Domain Articles uri icon

publication date

  • January 2022

start page

  • 218

end page

  • 238

issue

  • 2

volume

  • 10

International Standard Serial Number (ISSN)

  • 2227-7390

abstract

  • We study two seminal approaches, developed by B. Simon and J. Kisy´ nski, 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.

keywords

  • schrödinger equation; time-dependent hamiltonian; hilbert scales; time-dependent domain