Spectral fluctuations of Schrödinger operators generated by critical points of the potential

Details Spectral fluctuations of Schrödinger operators generated by critical points of the potential Spectral fluctuations of Schrödinger operators generated by critical points of the potential

2005-04-25

arxiv.org/abs/math-ph/0504073v2

Journal of Statistical Physics. Volume 123, Number 4 / May, 2006

Physics

Mathematical Physics

18 pages, Final version

Scientific paper

10.1007/s10955-006-9113-z

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.

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