Thierry Paul, Alejandro Uribe.
The semi-classical trace formula and propagation of
wave packets
(134K, LaTeX)
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 $\phi_t(\Lambda)$.