On the discriminant of Harper's equation

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.Comment: 10 pages, Latex, small change

Spectral estimates for periodic Jacobi matrices

We obtain bounds for the spectrum and for the total width of the spectral gaps for Jacobi matrices on $\ell^2(\Z)$ of the form $(H\psi)_n= a_{n-1}\psi_{n-1}+b_n\psi_n+a_n\psi_{n+1}$, where $a_n=a_{n+q}$ and $b_n=b_{n+q}$ are periodic sequences of real numbers. The results are based on a study of the quasimomentum $k(z)$ corresponding to $H$. We consider $k(z)$ as a conformal mapping in the complex plane. We obtain the trace identities which connect integrals of the Lyapunov exponent over the gaps with the normalised traces of powers of $H$.Comment: 18 pages, 5 figures, presentation improved, to appear in Commun. Math. Phy

Quasi-exactly solvable problems and the dual (q-)Hahn polynomials

A second-order differential (q-difference) eigenvalue equation is constructed whose solutions are generating functions of the dual (q-)Hahn polynomials. The fact is noticed that these generating functions are reduced to the (little q-)Jacobi polynomials, and implications of this for quasi-exactly solvable problems are studied. A connection with the Azbel-Hofstadter problem is indicated.Comment: 15 pages, LaTex; final version, presentation improved, title changed, to appear in J.Math.Phy