On a semiclassical formula for non-diagonal matrix elements

TL;DR

This paper rigorously proves a semiclassical asymptotic formula for non-diagonal matrix elements of bounded observables in the eigenbasis of Schrödinger operators with specific potential conditions, extending known physics results with mathematical rigor.

Contribution

It provides a rigorous mathematical proof of a semiclassical formula for non-diagonal matrix elements, previously known only in theoretical physics.

Findings

Asymptotic formula for matrix elements established

Rigorous mathematical proof provided

Applicable to Schrödinger operators with two turning points

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 eigen-energy of $H(\hbar)$. An asymptotic formula for $<{}n|W(x)|n+k>$, the non-diagonal 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.