 94274 T. Paul, A. Uribe
 On the pointwise behavior of semiclassical measures
(79K, LaTeX)
Aug 19, 94

Abstract ,
Paper (src),
View paper
(auto. generated ps),
Index
of related papers

Abstract. In this paper we concern ourselves with the small $\h$ asymptotics of
the inner products of the eigenfunctions of a Schr\"odingertype
operator with a coherent state. More precisely, let $\psi_j^\h$ and
$E_j^\h$ denote the eigenfunctions and eigenvalues of a
Schr\"odingertype operator $H_\h$ with discrete spectrum. Let
$\psi_{(x,\xi)}$ be a coherent state centered at the point $(x,\xi)$ in
phase space. We estimate as $\h\to 0$ the averages of the squares of
the inner products $ \mid(\psi_{(x,\xi)}^a,\psi_j^\hbar)\mid ^2 $ over
an energy interval of size $\h$ around a fixed energy, $E$. This
follows from asymptotic expansions of the form
\[
\sum_j\varphi\left( \frac{E_j(\hbar)E}{\hbar}\right)
\mid(\psi_{(x,\xi)}^a,\psi_j^\hbar)\mid ^2\
\sim \ \sum_{k=0}^\infty\, c_k(a) \hbar^{n+\frac{1}{2}+k}\,
\]
for certain test functions $\varphi$ and Schwartz amplitudes $a$ of the
coherent state. We compute the leading coefficient in the expansion,
which depends on whether the classical trajectory through $(x,\xi)$ is
periodic or not. In the periodic case the iterates of the trajectory
contribute to the leading coefficient. We also discuss the case of the
Laplacian on a compact Riemannian manifold.
 Files:
94274.tex