Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Spectral Asymptotics of the Two-Dimensional Schr\"odinger operator
  with a Strong Magnetic Field   We consider the spectral problem for the two-dimensional Schr\"odinger
operator for a charged particle in strong uniform magnetic and periodic
electric fields. The related classical problem is analyzed first by means of
the Krylov-Bogoljubov-Alfven and Neishtadt averaging methods. It allows us to
show ``almost integrability'' of the the original two-dimensional classical
Hamilton system, and to reduce it to a one-dimensional one on the phase space
which is a two-dimensional torus. Using the topological methods for integrable
Hamiltonian system and elementary facts from the Morse theory, we give a
general classification of the classical motion. According this classification
the classical motion is separated into different regimes with different
topological characteristics (like rotation numbers and Maslov indices). Using
these regimes, the semiclassical approximation, the Bohr-Sommerfeld rule and
the correspondence principle, we give a general asymptotic description of the
(band) spectrum of the original Schr\"odinger operator and, in particular,
estimation for the number of subbands in each Landau band. From this point of
view the regimes, are the classical preimages of ``spectral series'' of the
Schr\"odinger operator. We also discuss the relationship between this spectrum
and the spectrum of one-dimensional difference operators.

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The categories of this paper are: solv-int
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The categories of this paper are: bayes-an
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The categories of this paper are: q-fin
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The categories of this paper are: acc-phys
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The categories of this paper are: supr-con
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The categories of this paper are: alg-geom
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The categories of this paper are: chem-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: econ
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The categories of this paper are: hep-ex
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The categories of this paper are: nucl-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: cs
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The categories of this paper are: plasm-ph
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The categories of this paper are: physics
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The categories of this paper are: hep-th