Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 304

Phase Space Tunneling for Operators with Symbols in a Gevrey Class

K. Jung

Phase space tunneling and exponential decay of eigenfunctions in phase space are well known for operators with symbols which are analytic in some neighbourhood of the real axis. This can be used to proof an adiabatic theorem of exponential order if one assumes the Hamiltonian to depend analytically on time. However to study compactly supported switching processes one has to weaken the analyticity assumptions. Here we examine non-analytic symbols with Gevrey class regularity and show that we get an exponential decay of the corresponding eigenfunctions with respect to $hbar^{1/a}$ as $hbar$ tends to zero, where $a > 1$. The loss of regularity causes a slower decay in $hbar$. The analysis is done using the methods of A. Martinez and its generalization by S. Nakamura. An upper bound for the rate of decay is given.

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