Schroedinger Operator with Strong Magnetic Field near Boundary

We consider 2-dimensional Schroedinger operator with the non-degenerating magnetic field in the domain with the boundary and under certain non-degeneracy assumptions we derive spectral asymptotics with the remainder estimate better than $O(h^{-1})$, up to $O({\mu }^{-1}{h}^{-1})$ and the principal part $\asymp h^{-2}$where$h\ll 1$isPlanckconstantand$\mu \gg 1$ is the intensity of the magnetic field; $\mu h\le 1$. We also consider generalized Schr"odinger-Pauli operator in the same framework albeit with $\mu h\ge 1$ and derive spectral asymptotics with the remainder estimate up to O(1) and with the principal part $\asymp \mu h^{-1}$, or, under certain special circumstances with the principal part $\asymp {\mu }^{1\mathrm{/}2}{h}^{-1\mathrm{/}2}$.