Some inverse spectral results for semi-classical Schr\"odinger operators

TL;DR

This paper demonstrates that for a semi-classical Schrödinger operator with a potential having a unique global minimum and rationally independent Hessian eigenvalues, the low-lying eigenvalues uniquely determine the potential's Taylor series at the minimum.

Contribution

It establishes a novel inverse spectral result linking low-lying eigenvalues to the Taylor series of the potential in semi-classical Schrödinger operators.

Findings

Low-lying eigenvalues determine the Taylor series of the potential.

Unique determination of the potential's local behavior from spectral data.

Conditions on the Hessian eigenvalues ensure the inverse result.

Abstract

We consider a semi-classical Schr\"odinger operator, -h^2\Delta + V(x). Assuming that the potential admits a unique global minimum and that the eigenvalues of the Hessian are linearly independent over the rationals, we show that the low-lying eigenvalues of the operator determine the Taylor series of the potential at the minimum.