On the Schr\"odinger Equation for Time-Dependent Hamiltonians with a Constant Form Domain

TL;DR

This paper compares two foundational methods for establishing the well-posedness of the Schrödinger equation with time-dependent Hamiltonians that have a constant form domain but possibly variable operator domain, providing a detailed analysis of their assumptions.

Contribution

It offers a comprehensive comparison of Simon's and Kisyński's approaches, clarifying their assumptions and connecting sesquilinear forms with bounded operators in the context of time-dependent Schrödinger equations.

Findings

Characterization of continuity and differentiability of form- and operator-valued functions

Extensive comparison of the two approaches and their assumptions

Clarification of the relation between sesquilinear forms and bounded operators

Abstract

We study two seminal approaches, developed by B. Simon and J. Kisy\'nski, to the well-posedness of the Schr\"odinger equation with a time-dependent Hamiltonian. In both cases the Hamiltonian is assumed to be semibounded from below and to have 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 characterization of continuity and differentiability properties of form-valued and operator-valued functions which enables an extensive comparison between the two approaches and their technical assumptions.