On a Semiclassical Formula for Non-Diagonal Matrix Elements

Abstract

Let H(ℏ)=−ℏ 2d2/dx 2+V(x) be a Schrodinger 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 ∞=lim inf |x|→∞ V(x). We consider the semiclassical limit n→∞, ℏ=ℏ n →0 and E n =E where E n is the nth eigenenergy of H(ℏ). An asymptotic formula for 〈n|W(x)|n+k〉, the non-diagonal matrix elements of W(x) in the eigenbasis of H(ℏ), has been known in the theoretical physics for a long time. Here it is proved in a mathematically rigorous manner.