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The asymptotic form of the lower Landau bands in a strong magnetic field. (English. Russian original) Zbl 1039.81023
Theor. Math. Phys. 131, No. 2, 704-728 (2002); translation from Teor. Mat. Fiz. 131, No. 2, 304-331 (2002).
Summary: The asymptotic form of the bottom part of the spectrum of the two-dimensional magnetic Schrödinger operator with a periodic potential in a strong magnetic field is studied in the semiclassical approximation. Averaging methods permit reducing the corresponding classical problem to a one-dimensional problem on the torus; we thus show the “almost integrability” of the original problem. Using elementary corollaries from the topological theory of Hamiltonian systems, we classify the almost invariant manifolds of the classical Hamiltonian. The manifolds corresponding to the bottom part of the spectrum are closed or nonclosed curves and points. Their geometric and topological characteristics determine the asymptotic form of parts of the spectrum (spectral series). We construct this asymptotic form using the methods of the semiclassical approximation with complex phases. We discuss the relation of the asymptotic form obtained to the magneto-Bloch conditions and asymptotics of the band spectrum.

MSC:
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
82D20 Statistical mechanical studies of solids
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