Year: 2013
Type: preprint
Abstract: Starting from the semi-classical spectrum of Schrödinger operators $-h^2Δ+V$ (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; Schrödinger operators; Inverse spectral problems.
Source: arXiv (Cornell University)
Author: Camus, Brice
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Deutsche Forschungsgemeinschaft
OpenAlex ID: W2468838125