On the discriminant of Harper's equation

Details On the discriminant of Harper's equation On the discriminant of Harper's equation

1999-09-17

arxiv.org/abs/math-ph/9909021v2

Lett.Math.Phys. 52 (2000) 155-163

Physics

Mathematical Physics

10 pages, Latex, small changes

Scientific paper

The spectrum of Harper's equation is determined by the discriminant, which is a certain polynomial of degree Q if the commensurability parameter of Harper's equation is P/Q, where P, Q are coprime positive integers. A simple expression is indicated for the derivative of the discriminant at zero energy for odd Q. Three dominant terms of the asymptotics of this derivative are calculated for the case of an arbitrary P as Q increases. The result gives a lower bound on the width of the centermost band of Harper's equation and shows the effects of band clustering. It is noticed that the Hausdorff dimension of the spectrum is zero for the case P=1, Q infinitely large.

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