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The Harper equation describes an electron on a 2D lattice in magnetic fieldand a particle on a 1D lattice in a periodic potential, in general,incommensurate with the lattice potential. We find the distribution of theroots of Bethe ansatz equations associated with the Harper equation in thelimit as alpha=1/Q tends to 0, where alpha is the commensurability parameter (Qis integer). Using the knowledge of this distribution we calculate the higherand lower boundaries of the spectrum of the Harper equation for small alpha.The result is in agreement with the semiclassical argument, which can be usedfor small alpha.Comment: 17 pages including 5 postscript figures, Latex, minor changes, to  appear in Phys.Rev.

Bethe ansatz for the Harper equation: Solution for a small commensurability parameter