Adiabatic theorems for general linear operators with time-independent domains

TL;DR

This paper proves new adiabatic theorems for general linear operators with fixed domains, applicable to dissipative systems, without requiring spectral values to be semisimple, and under weak regularity conditions, with applications in quantum systems.

Contribution

It extends adiabatic theorems to broader classes of operators without spectral gap or semisimplicity requirements, with minimal regularity assumptions.

Findings

Established adiabatic theorems without spectral gap condition

Applicable to dissipative and time-varying systems

Demonstrated relevance to quantum systems and switching processes

Abstract

We establish adiabatic theorems with and without spectral gap condition for general -- typically dissipative -- linear operators $A(t): D(A(t)) \subset X \to X$ with time-independent domains $D(A(t)) = D$ in some Banach space $X$. Compared to the previously known adiabatic theorems -- especially those without spectral gap condition -- we do not require the considered spectral values $\lambda(t)$ of $A(t)$ to be (weakly) semisimple. We also impose only fairly weak regularity conditions. Applications are given to slowly time-varying open quantum systems and to adiabatic switching processes.