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Spectrum of a two‐dimensional Harper's operator

We give a semiclassical study of the spectrum of a two-dimensional model of Harper's operator. We localize the part of the spectrum near the maximum in exponentially small intervals and give estimate on the splitting between them. Then we prove that in the rational case the Lebesgue measure of the spectrum remains strictly positive even in the semiclassical limit. Some spectral problems in solid state physics can be reduced to the study of finite difference equations. Harper's equation is a famous one in this direction. It is a model for the Bloch electron lying in a two- dimensional lattice under a periodic magnetic field. The periodicity and symmetry properties of these equations are consequence of those of the magnetic and electric fields in the Schrodinger models. Using the semiclassical Weyl quantization the difference equations should be viewed as realizations of some pseudodifferential operators on the Hilbert space L 2 and whose symbols have the same periodicities. Harper's operator is the semiclassical quantization of the periodic symbol cos ξ + cos x and acting on L 2 (R )b y