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Semiclassical electron motion and Novikov’s conjecture. (English. Russian original) Zbl 0932.58039

J. Math. Sci., New York 94, No. 4, 1589-1592 (1999); translation from Zap. Nauchn. Semin. POMI 235, 228-234 (1996).
Summary: The structure of nonclosed trajectories of semiclassical electron motion in a crystal in a weak constant and uniform magnetic field of irrationality degree 3 is considered. It is proved that two cases can exist. In the first case, the set of energy levels which contain nonclosed trajectories is a closed interval and any regular nonclosed trajectory lies in a finite-wide stripe and comes through it in one direction. In the order case, there is only one energy level containing nonclosed trajectories.

MSC:

58J90 Applications of PDEs on manifolds
78A35 Motion of charged particles

Citations:

Zbl 0924.00015

References:

[1] S. P. Novikov, ”The Hamiltonian formalism and multivalued analogue of Morse theory,”Usp. Mat. Nauk,37, No. 5, 3–49 (1982).
[2] S. P. Novikov, ”Critical points and level surfaces of multivalued functions,”Proc. of Steklov Institute of Mathematics,166, 201–209 (1984). · Zbl 0553.58005
[3] A. V. Zorich, ”Novikov problem about semiclassical electron motion in a uniform magnetic field close to rational,”Usp. Mat. Nauk,39, 235–236 (1984).
[4] I. A. Dynnikov, ”The proof of the Novikov conjecture in the case of small disturbances of rational magnetic fields,”Usp. Mat. Nauk,47, 161–162 (1992). · Zbl 0778.58016
[5] S. P. Novikov, ”Quasiperiodic Structures in Topology,” in:Proc. Congress Topological Methods in Mathematics, Stony Brook, June (1991). · Zbl 0813.57029
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