On spectral properties of Harper-like models

We study spectral properties of Harper-like models by algebraic and combina torial methods and derive sufficient conditions for the existence of spectr al gaps with qualitative estimates. For this class the Chambers relation ho lds and we obtain an analytic expression for the representation dependent p art. Models corresponding to the rectangular and triangular lattice are stu died. In the second case we show that one class of spectral gaps is open fo r magnetic fields with "rational magnetic flux per unit cell.'' A quantitat ive estimate for the gap widths is given for the anisotropic case and for " irrational magnetic flux'' fulfilling some Liouville condition the spectrum is a Cantor set. (C) 1999 American Institute of Physics. [S0022-2488(99)03 401-5].