Propagators weakly associated to a family of Hamiltonians and the adiabatic theorem for the Landau Hamiltonian with a time-dependent Aharonov-Bohm flux

TL;DR

This paper develops a new notion of propagator weakly associated to time-dependent Hamiltonians and proves an adiabatic theorem for a quantum particle in a magnetic field with a time-dependent Aharonov-Bohm flux, addressing models with changing domains.

Contribution

It introduces the concept of a propagator weakly associated to a family of Hamiltonians and establishes an adiabatic theorem for such systems with time-dependent domains.

Findings

Defined and developed the notion of a propagator weakly associated to time-dependent Hamiltonians.

Proved an adiabatic theorem for quantum systems with a time-dependent Aharonov-Bohm flux.

Extended adiabatic results to models with singular flux tubes and changing Hamiltonian domains.

Abstract

We study the dynamics of a quantum particle moving in a plane under the influence of a constant magnetic field and driven by a slowly time-dependent singular flux tube through a puncture. The known adiabatic results do not cover these models as the Hamiltonian has time dependent domain. We give a meaning to the propagator and prove an adiabatic theorem. To this end we introduce and develop the new notion of a propagator weakly associated to a time-dependent Hamiltonian.