On the Schrödinger Equation for Time-Dependent Hamiltonians with a Constant Form Domain

Abstract

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.

Quantum controllability on graph-like manifolds through magnetic potentials and boundary conditions

“…The main obstruction is the fact that the controls obtained by applying theorem 5.4 are piecewise constant. If they were smooth, the results in [BLP22b,Kis64] would ensure that the piecewise weak solutions are also strong solutions. A first attempt to get approximate controllability with strong solutions of the Schrödinger equation would be to use the stability results developed in [Bal21] to extend theorem 2.8 to the case of smooth controls.…”

Quantum mechanics on time-varying space domains

“…It will be the Dirichlet problem we are concerned with in the present paper. With all of this interest in how moving boundaries change the dynamics of quantum mechanical systems, there has also been a recent interest in both understanding observability in these systems [27] as well as using moving boundaries to control observables [28,29]. A generic treatment of the question of controlling observables via control of a moving boundary was carried out in [30].…”