94-159 Thierry Paul, Alejandro Uribe.

The semi-classical trace formula and propagation of wave packets (134K, LaTeX) Jun 1, 94

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Abstract. We study spectral and propagation properties of operators of the form $S_\h = \sum_{j=0}^N \h^j P_j$ where $\forall j$ $P_j$ is a differential operator of order $j$ on a manifold $M$, asymptotically as $\h\to 0$. The estimates are in terms of the flow $\{\phi_t\}$ of the classical Hamiltonian $H(x,p) = \sum_{j=0}^N \sigma_{P_j}(x,p)$ on $T^*M$, where $\sigma_{P_j}$ is the principal symbol of $P_j$. We present two sets of results. (I) The ``semiclassical trace formula", on the asymptotic behavior of eigenvalues and eigenfunctions of $S_\h$ in terms of periodic trajectories of $H$. (II) Associated to certain isotropic submanifolds $\Lambda\subset T^*M$ we define families of functions $\{\psi_\h\}$ and prove that $\forall t$ $\{\exp(-it\h S_h)(\psi_\h )\}$ is a family of the same kind associated to $\phi_t(\Lambda)$.

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