 01340 Yu. Karpeshina
 System of Model Functions for the TwoDimensional Periodic Magnetic Schr\"{o}dinger Operator.
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Sep 25, 01

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Abstract. For the periodic magnetic Schr\"{o}dinger operator
in two dimensions
we describe
a set of model functions,
which solve the equations for eigenfunctions approximately . There exist the model functions
of two types: a weak diffraction type and a strong diffraction
type. It is shown that the model functions are mutually ``almost orthogonal" and
the model set is complete in the high
energy region  all eigenfunctions with eigenvalues large enough
can be described in terms of the model functions. The present paper contains the construction
of the system. This is the first of two papers designed to prove that in the high
energy region each
eigenfunction is close to exactly one of the model functions for
a rich set of quasimomenta, for the rest of quasimomenta it is
close to a linear combination of the model functions. Information
about the isoenergetic surface, a proof of the BetheSommerfeld
conjecture and an asymptotic of the integral density of states in
the high energy region are going to be obtained as corollaries of the formulae for
eigenfunctions.
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