Semiclassical Spectral Invariants for Schr\"odinger Operators

TL;DR

This paper investigates semiclassical spectral measures for Schrödinger operators, computes their asymptotic coefficients, and applies these results to potential recovery and magnetic field effects, linking spectral data to classical normal forms.

Contribution

It provides explicit asymptotic coefficients for spectral measures and extends potential recovery methods to magnetic Schrödinger operators.

Findings

Computed first coefficients of spectral measure asymptotics.

Provided an alternative proof for potential recovery from spectral data.

Generalized asymptotic expansions to magnetic field cases.

Abstract

In this article we study the semiclassical spectral measures associated with Schr\"odinger operators on $R^n$. In particular we compute the first few coefficients of the asymptotic expansions of these measures and, as an application, give an alternative proof of Colin de Verdiere's result on recovering one dimensional potential wells from semiclassical spectral data. We also study the relation of semiclassical spectra measures to Birkhoff normal forms and describe a generalization of the asymptotic expansion above to the Schr\"odinger operator in the presence of a magnetic field.