Spectral fluctuations of Schr\"odinger operators generated by critical points of the potential

TL;DR

This paper introduces a semi-classical method based on spectral analysis to detect and partially reconstruct critical points of a potential in Schr"odinger operators, linking spectral fluctuations to classical equilibrium points.

Contribution

It develops a mathematically rigorous approach using Gutzwiller's trace formula to identify potential critical points from spectral data, advancing spectral analysis techniques.

Findings

Spectral fluctuations reveal critical points of the potential.

The method allows partial reconstruction of the potential's local shape.

Generalizations of the approach extend its applicability.

Abstract

Starting from the spectrum of Schr\"odinger operators on $\mathbb{R}^n$, we propose a method to detect critical points of the potential. We argue semi-classically on the basis of a mathematically rigorous version of Gutzwiller's trace formula which expresses spectral statistics in term of classical orbits. A critical point of the potential with zero momentum is an equilibrium of the flow and generates certain singularities in the spectrum. Via sharp spectral estimates, this fluctuation indicates the presence of a critical point and allows to reconstruct partially the local shape of the potential. Some generalizations of this approach are also proposed.\medskip keywords : Semi-classical analysis; Schr\"odinger operators; Equilibriums in classical mechanics.