The Adiabatic Theorem for SwitchingProcesses with Gevrey Class Regularity

The adiabatic theorem in quantum mechanics can be understood as an eeect of phase space tunneling 12]. This allows us to use micro-local methods from the theory of pseudo diierential operators to proof an adiabatic theorem where the Hamiltonian (as a function of time) belongs to some Gevrey class. These classes are of interest since they can be used to model a compactly supported and smooth switching process. A reened deenition of Gevrey classes and an optimized almost analytic extension is introduced. They are used to proof an adiabatic theorem where the decay of the adiabatic invariant is exponentially small w.r.t. to " 1=a , a > 1, where " is the adiabatic parameter. The rate of decay is explicitly given as a function of the parameters that specify the smoothness of the considered Gevrey class.