On a semiclassical formula for non-diagonal matrix elements

Abstract

Let $H(\hslash )=-{\hslash }^2{d}^2/d{x}^2+V(x)$ be a Schrödinger 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 $V_{\infty }={\mathrm{liminf}}_{|x|\to \infty }V(x)$. We consider the semiclassical limit $n\to \infty$, $\hslash ={\hslash }_n\to 0$ and $E_n=E$ where $E_n$ is the $n$th eigen-energy of $H(\hslash )$. An asymptotic formula for $<n|W(x)|n+k>$, the non-diagonal matrix elements of $W(x)$ in the eigenbasis of $H(\hslash )$, has been known in the theoretical physics for a long time. Here it is proved in a mathematically rigorous manner.

LaTeX2e