Adiabatic theorems for general linear operators with time-dependent domains

TL;DR

This paper develops adiabatic theorems for general linear operators with time-dependent domains in Banach spaces, including dissipative and skew-adjoint operators, without requiring spectral values to be semisimple.

Contribution

It generalizes existing adiabatic theorems to broader classes of operators with time-dependent domains, removing the semisimplicity requirement.

Findings

Established adiabatic theorems without spectral gap conditions.

Applied the theorems to skew-adjoint operators from symmetric forms.

Extended the known adiabatic theorem to operators with time-dependent domains.

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-dependent domains $D(A(t))$ in some Banach space $X$. In these theorems, we do not require the considered spectral values $\lambda(t)$ of $A(t)$ to be (weakly) semisimple. We then apply our general theorems to the special case of skew-adjoint operators $A(t) = 1/i A_{a(t)}$ defined by symmetric sesquilinear forms $a(t)$ and thus generalize, in a very simple way, the only adiabatic theorem for operators with time-dependent domains known so far.