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 Dirac Energy. II. Magnetic Schroedinger operator
I derive sharp semiclassical asymptotics of \int |e_h(x,y,0)|^2\omega (x,y)dxdy where e_h(x,y,\tau) is the Schwartz kernel of the spectral projector of Magnetic Schroedinger operator and \omega (x,y) is singular as x=y. I also consider asymptotics of more general expressions.
Sharp Spectral Asymptotics for four-dimensional Schroedinger operator with a strong magnetic field. II
I consider 4-dimensional Schr\"odinger operator with the generic non-degenerating magnetic field and for a generic potential I derive spectral asymptotics with the remainder estimate $O(\mu^{-1}h^{-3})$ and the principal part $\asymp h^{-4}$ where $h\ll 1$ is Planck constant and $\mu \gg 1$ is the intensity of the magnetic field. For general potentials remainder estimate $O(\mu^{-1}h^{-3}+\mu^2h^{-2})$ is achieved.