 06317 O. Lev, P. Stovicek
 On a semiclassical formula for nondiagonal matrix elements
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Nov 6, 06

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Abstract. Let $H(\hbar)=\hbar^2d^2/dx^2+V(x)$ be a Schr\"odinger operator on the real line, $W(x)$ be a bounded observable depending only on the coordinate and $k$ be a fixed integer. Suppose that an energy level $E$ intersects the potential $V(x)$ in exactly two turning points and lies below $V_\infty=\liminf_{x\to\infty}\,V(x)$. We consider the semiclassical limit $n\to\infty$, $\hbar=\hbar_n\to0$ and $E_n=E$ where $E_n$ is the $n$th eigenenergy of $H(\hbar)$. An asymptotic formula for $\langle{}nW(x)n+k\rangle$, the nondiagonal matrix elements of $W(x)$ in the eigenbasis of $H(\hbar)$, has been known in the theoretical physics for a long time. Here it is proved in a mathematically rigorous manner.
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