Abstract
We consider a Schrödinger operator of a charge motion near a surface in a strong homogeneous external magnetic field in the resonant case when the cyclotron frequency is equal to the frequency of the quadratic confinement potential holding the charge near the surface. The small curvature of the surface causes a significant energy splitting of the Landau levels in this case. The corresponding drift of the charge oscillations centers induces a geometric current on the surface. Using the quantum averaging over the cyclotron and confinement oscillations we reduce the \(3\)-dimensional magnetic Schrödinger operator to an effective \(2\)-dimensional operator, which determines the small splittings of Landau levels and the quantization of corresponding current on the surface. We obtain an explicit form of the second order correction induced by the surface curvature to the lowest Landau level near a nondegenerate stationary point of the surface and the corresponding closed curves of the geometric current on the surface.
DOI 10.1134/S1061920822040185
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Acknowledgments
The author is sincerely grateful to M. Karasev and G. Safyanov with whom this problem was discussed at the early stages.
Funding
This research was supported by the Russian Science Foundation under grant No. 19-71-10003.
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Vybornyi, E.V. Magneto-Dimensional Resonance on Curved Surfaces. Russ. J. Math. Phys. 29, 595–600 (2022). https://doi.org/10.1134/S1061920822040185
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DOI: https://doi.org/10.1134/S1061920822040185