Semiclassical Spectral Invariants for Schrödinger Operators

Abstract

In this article we show how to compute the semiclassical spectral measure associated with the Schrödinger operator on $\mathbb{R}^n$, and, by examining the first few terms in the asymptotic expansion of this measure, obtain inverse spectral results in one and two dimensions. (In particular we show that for the Schrödinger operator on $\mathbb{R}^2$ with a radially symmetric electric potential, $V$, and magnetic potential, $B$, both $V$ and $B$ are spectrally determined.) We also show that in one dimension there is a very simple explicit identity relating the spectral measure of the Schrödinger operator with its Birkhoff canonical form.