2021/12/31 by Aitor Balmaseda, Davide Lonigro, Juan Manuel Pérez-Pardo · 5 citations
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Bounded function #Computer science #Connection (principal bundle) #Constant (computer programming) #Differentiable function #Domain (mathematical analysis) #Hamiltonian (control theory) #Hermitian matrix #Mathematical analysis #Mathematical optimization #Mathematical physics #Mathematics #Numerical methods for differential equations #Operator (biology) #Pure mathematics #Sesquilinear form #Spectral Theory in Mathematical Physics #math-ph #math.FA #math.MP #msc:35J10 #msc:35Q41 #msc:46N50 #msc:47B25 #msc:81Q10
paper · pdf · doi:10.3390/math10020218
published in Mathematics 10(2), 218 (Multidisciplinary Digital Publishing Institute)
arxiv created 2022/01/11 · openalex publication_date 2022/01/11 · arxiv updated 2022/01/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.