Adiabatic theorems with and without spectral gap condition for non-semisimple spectral values

Abstract

We establish adiabatic theorems with and without spectral gap condition for general operators $A(t):D(A(t))\subset X\to X$ with possibly time-dependent domains in a Banach space $X$. We first prove adiabatic theorems with uniform and non-uniform spectral gap condition (including a slightly extended adiabatic theorem of higher order). In these adiabatic theorems the considered spectral subsets $\ddot{I}ƒ(t)$ have only to be compact -- in particular, they need not consist of eigenvalues. We then prove an adiabatic theorem without spectral gap condition for not necessarily (weakly) semisimple eigenvalues: in essence, it is only required there that the considered spectral subsets $\ddot{I}ƒ(t)=\{\hat{I}»(t)\}$ consist of eigenvalues $\hat{I}»(t)\in \partial \ddot{I}ƒ(A(t))$ and that there exist projections $P(t)$ reducing $A(t)$ such that $A(t){|}_{P(t)D(A(t))}-\hat{I}»(t)$ is nilpotent and $A(t){|}_{(1-P(t))D(A(t))}-\hat{I}»(t)$ is injective with dense range in $(1-P(t))X$ for almost every~$t$. In all these theorems, the regularity conditions imposed on $t\mapsto A(t)$, $\ddot{I}ƒ(t)$, $P(t)$ are fairly mild. We explore the strength of the presented adiabatic theorems in numerous examples. And finally, we apply the adiabatic theorems for time-dependent domains to obtain -- in a very simple way -- adiabatic theorems for operators $A(t)$ defined by symmetric sesquilinear forms.