Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 442

The Adiabatic Theorem for Switching Processes with Gevrey Class Regularity

Klaus Jung

The adiabatic theorem in quantum mechanics can be understood as an effect of phase space tunneling. This allows us to use microlocal methods from the theory of pseudo differential 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 refined definition 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 $epsilon^{1/a}$, $a>1$, where $epsilon$ 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.

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