Quasiclassical approximations for almost-Mathieu equations

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Abstract
Quasiclassical approximations are used to investigate the edge of the spectrum of almost-Mathieu equations. Two cases are worked in details : the tight-binding model with an applied magnetic field on a square and triangular lattices. It is shown that both approaches lead to identical results. New results, relating the density of states to the shape (local slopes) of the spectrum edge are obtained. In general, close to commensurability, the energy levels exhibit an asymmetry which is a consequence of the existence of a phase holonomy (adiabatic or Berry's phase) in the WKB theory. The possibility of a direct experimental manifestation of this asymmetry, on the magnetic properties of the mixed state of a superconducting network is discussed