Sharp Spectral Asymptotics for 2-dimensional Schr\"odinger operator with a strong magnetic field. Note about forgotten generic case
I consider magnetic Schr\"odinger operator in dimension $d=2$ assuming that coefficients are smooth and magnetic field is non-degenerating. Then I extend the remainder estimate $O(\mu^{-1}h^{-1}+1)$ derived in \cite{Ivr1} for the case when $V/F$ has no stationary points to the case when it has non-degenerating stationary points. If some of them are saddles and $\mu^3h\ge 2$ then asymptotics contains correction terms of magnitude $\mu^{-1}h^{-1}|\log \mu^3 h|$.
Sharp Spectral Asymptotics for two-dimensional Schrödinger operator with a strong degenerating magnetic field
I consider two-dimensional Schr\"odinger operator with degenerating magnetic field and in the generic situation I derive spectral asymptotics as $h\to +0$ and $\mu\to +\infty$ where $h$ and $\mu$ are Planck and coupling parameters respectively. The remainder estimate is $O(\mu^{-{\frac 1 2}}h^{-1})$ which is between $O(\mu^{-1}h^{-1})$ valid as magnetic field non-degenerates and $O(h^{-1})$ valid as magnetic field is identically 0. As $\mu$ is close to its maximal reasonable value $O(h^{-2})$ the principal part contains correction terms associated with short periodic trajectories of the corresponding classical dynamics.
PHASE TRANSITION IN THE SEMICLASSICAL REGIME
In continuation with [7] and an unpublished paper by F. Daumer [4], we would like to explore the variation of the splitting for a Schrodinger operator coming from field theory when the potential changes as a function of the temperature from a one-well situation to a double well situation. We are for the moment only able to treat the fixed dimension case. But, because the most interesting thing would be to analyze the dependence on the dimension (in the perspective of applications to statistical mechanics) we have tried to follow some of the steps in the proof with respect to the dimension. All this analysis was strongly motivated by a course by M. Kac where some heuristical discussion based on some Born–Oppenheimer analysis is presented. So our work can be considered as a first attempt for justifying this discussion.