A semi-classical inverse problem I: Taylor expansions

TL;DR

This paper demonstrates that in one dimension, the Taylor expansion of a potential around a non-degenerate critical point can be reconstructed from the semi-classical spectrum of the Schrödinger operator near the critical value.

Contribution

It introduces a method to recover the Taylor expansion of potentials from spectral data in a semi-classical setting for one-dimensional Schrödinger operators.

Findings

Potential Taylor expansions are uniquely determined by semi-classical spectra.

The method applies near non-degenerate critical points.

Spectral data near the critical value suffices for reconstruction.

Abstract

In dimension 1, we show that the Taylor expansion of a potential near a generic non degenerate critical point can be recovered from the knowledge of the semi-classical spectrum of the associated Schr\"odinger operator near the corresponding critical value.