Inverse spectral problems for Schr\"odinger and pseudo-differential operators

TL;DR

This paper explores how semi-classical spectral data of Schr"odinger operators can be used to identify and reconstruct critical points and local features of the potential function using microlocal analysis techniques.

Contribution

It introduces methods to detect critical energy levels and reconstruct local potential shapes from spectral invariants, extending previous inverse spectral approaches.

Findings

Spectral invariants reveal critical points of the potential.

Singularities in eigenvalue distributions indicate equilibrium states.

Methods allow partial reconstruction of potential near critical points.

Abstract

Starting from the semi-classical spectrum of Schr\"odinger operators $-h^2\Delta+V$ (on $\mathbb{R}^n$ or on a Riemannian manifold) it is possible to detect critical levels of the potential $V$. Via micro-local methods one can express spectral statistics in terms of different invariants: \begin{itemize} \item Geometry of energy surfaces (heat invariant like). \item Classical orbits (wave invariants). \item But also classical equilibria (new wave invariants). \end{itemize} Any critical point of $V$ with zero momentum is an equilibrium of the flow and generates many singularities in the semi-classical distribution of eigenvalues. Via sharp spectral estimates, this phenomena indicates the presence of a critical energy level and the information contained in this singularity allows to reconstruct partially the local shape of $V$. Several generalizations of this approach are also proposed. Keywords : Spectral analysis, P.D.E., Micro-local analysis; Schr\"odinger operators; Inverse spectral problems.