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ABSTRACT:
The celebrated Harper equation deals
with the description of Bloch electrons on a two dimensional lattice
threaded by a transversal and homogeneous magnetic field. This results in
a second order discrete equation, which can be solved with the help of
three term recurrence relations. One accounts for the influence of
both commensurability ( i.e. of the number of magnetic flux quanta per
unit cell) and anisotropy parameters. We shall restrict ourselves to
rational values of the commensurability parameter like P/Q, where P and Q
are mutually prime integers. This yields Q-degree energy polynomials, for
which the wavefunction is the generating function. Having obtained
such polynomials opens the way to the derivation of the energy bands as
well as of the density of states. A q-symmetrized version of the
Harper-equation, which serves to the middle band description, is also
discussed in some detail. To this aim the q-calculus can be applied in a
quite efficient manner.
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