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Spectre de l’opérateur de Schrödinger magnétique avec symétrie d’ordre six. (Spectrum of the magnetic Schrödinger operator with symmetry of order six). (French) Zbl 0769.35075
This memoir is devoted to the semi-classical study of a two dimensional Schrödinger operator with periodic magnetic field and electric potential on a lattice with sixfold rotational symmetry. In the same spirit as in the memoirs of B. Helffer and J. Sjöstrand concerning the Harper’s equation [ibid. 34 (1988; Zbl 0714.34130)] the author treats the so-called triangular and hexagonal lattices, with the assumption that the electric potential \(V\) has one or two nondegenerate minima per periodicity cell.
The problem is reduced to the study of an effective Hamiltonian which appears to be a pseudodifferential operator on \(L^ 2(\mathbb{R})\) (or a system of pseudo-differential operators) whose symbol appears to be periodic with respect to the two variables in \(\mathbb{R}^ 2\) and have again some sixfold symmetry. With some assumptions on the flux and the studied energy level, the author can pursue the investigation and gives elements suggesting Cantor structure for the spectrum in agreement with some results obtained by physicists.
Reviewer: B.Helffer (Paris)

MSC:
35S05 Pseudodifferential operators as generalizations of partial differential operators
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
35J10 Schrödinger operator, Schrödinger equation
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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