An explicitly solvable model with a periodic system
of the Aharonov-Bohm vorteces
V.A.Geyler, A.V.Popov
The spectral analysis of the periodic Schrodinger operators
with a uniform magnetic field is an extremely complicated
problem, which is unsolved up to now. That is why various
approximate discrete models of the periodic systems in a
magnetic field are widely used. However the purely discrete
models are useless for investigation of periodic arrays of
quantum dots or quantum antidots. In this connection, the
lattice models with an "internal structure" are proposed
for periodic systems [1], [2]. In the case of a uniform
magnetic field both theoretical and numerical study of
the model Hamiltonian for the squre lattice reduces to the
study of the well known discrete Harper operator [3], [4],
etc. A fractal structure of the "flux-energy" diagramm
can be considered as a manifestation of the chaotic dynamics
in this case.
In our report some results of a theoretical
and numerical analysis of lattice models with an internal
structure are presented in the presence a
periodic system of Aharonov-Bohm vorteces. We consider
the cases of the square as well as honeycomb lattices.
[1] V.A.Geyler, B.S.Pavlov, I.Yu.Popov.
J. Math. Phys. 37 (1996), 5171 - 5194.
[2] V.A.Geyler, B.S.Pavlov, I.Yu.Popov.
Atti Sem. Mat. Fis. Univ. Modena. 45 (1997), 1-46.
[3] V.A.Geyler, I.Yu.Popov. Z. Phys. B98 (1995),473 - 477.
[4] V.A.Geyler, A.V.Popov. Pitman Res. Notes Math. 374 (1997), 74 - 78.