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Article

Averaging and spectral bands for 2-D magnetic Schrödinger operator with the growing and one-direction periodic potential

Russian Journal of Mathematical Physics. 2019. Vol. 26. No. 3. P. 265-276.
Anikin A., Brüning J., Dobrokhotov S., Vybornyi E.

In this paper, we consider the spectral problem for the magnetic Schrödinger operator on the 2-D plane (x1, x2) with the constant magnetic field normal to this plane and with the potential V having the form of a harmonic oscillator in the direction x1 and periodic with respect to variable x2. Such a potential can be used for modeling a long molecule. We assume that the magnetic field is quite large, this allows us to make the averaging and to reduce the original problem to a spectral problem for a 1-D Schrödinger operator with effective periodic potential. Then we use semiclassical analysis to construct the band spectrum of this reduced operator, as well as that of the original 2-D magnetic Schrödinger operator.