Total bandwidth for the Harper equation. III. Corrections to scaling

Earlier work on the spectral measure for the Harper equation (discrete Mathieu equation) showed that, if there is a large common period p for the lattice and the sinusoidal potential, the spectral measure scaled as C/p, where C is a constant independent of the number of oscillations q of the potential in p lattice spacings. The corrections to this scaling law are found for q=1 and q=2. For q=1 it is shown that the corrections to scaling are logarithmic functions of p, while for q=2 the leading corrections are reduced by a factor p -2 . Analytical and numerical support is given for the assertion that this difference between odd and even values of q persists for higher values of q provided they are substantially less than p.