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

Aitor Balmaseda,Davide Lonigro,Juan Manuel Pérez-Pardo

doi:10.3390/math10020218

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.

Highlights

A quantum dynamical system is a first-order linear evolution equation on a separable, complex Hilbert space H, where the evolution is determined by a family of densely defined, self-adjoint operators {H(t)}t∈I, where I ⊂ R is an interval
The dynamics is given by the time-dependent Schrödinger equation: d dt
Since we did not assume any specific expression for the Hamiltonian H(t), our work may provide a useful reference for mathematical and theoretical physicists interested in such problems

Summary

Introduction

A quantum dynamical system is a first-order linear evolution equation on a separable, complex Hilbert space H, where the evolution is determined by a family of densely defined, self-adjoint operators {H(t)}t∈I, where I ⊂ R is an interval. The aim of this work is to revise, in a common language, both approaches to the wellposedness problem for the non-autonomous Schrödinger equation with a constant form domain and (possibly) non-constant operator domain and to clarify the connections between them. Since we did not assume any specific expression for the Hamiltonian H(t), our work may provide a useful reference for mathematical and theoretical physicists interested in such problems This may inspire, for example, new developments in the study of quantum Hamiltonians with time-dependent boundary conditions. (ii) h(Ψ, Φ) = Ψ, TΦ for any Ψ ∈ dom h, Φ ∈ D; (iii) D is a core for h, that is D · h = dom h Note that this theorem establishes a one-to-one correspondence between closed, semibounded Hermitian sesquilinear forms and semibounded self-adjoint operators and motivates the following definition. H is called the sesquilinear form represented by T

Scales of Hilbert Spaces

Regularity of Operator-Valued and Form-Valued Functions

Example

Conclusions