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Let $H(\hbar)=-\hbar^2d^2/dx^2+V(x)$ be a Schr\"odinger operator on the realline, $W(x)$ 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 $n$theigen-energy of $H(\hbar)$. An asymptotic formula for $<{}n|W(x)|n+k>$, thenon-diagonal matrix elements of $W(x)$ in the eigenbasis of $H(\hbar)$, hasbeen known in the theoretical physics for a long time. Here it is proved in amathematically rigorous manner.Comment: LaTeX2

On a Semiclassical Formula for Non-Diagonal Matrix Elements