  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.
