01-340 Yu. Karpeshina

System of Model Functions for the Two-Dimensional Periodic Magnetic Schr\"{o}dinger Operator. (946K, postscript) 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 Bethe-Sommerfeld 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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