94-274 T. Paul, A. Uribe

On the pointwise behavior of semi-classical measures (79K, LaTeX) Aug 19, 94

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Abstract. In this paper we concern ourselves with the small $\h$ asymptotics of the inner products of the eigenfunctions of a Schr\"odinger-type operator with a coherent state. More precisely, let $\psi_j^\h$ and $E_j^\h$ denote the eigenfunctions and eigenvalues of a Schr\"odinger-type 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.

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