Central spectral gaps of the almost Mathieu operator

We consider the spectrum of the almost Mathieu operator $H_\alpha$ with frequency $\alpha$ and in the case of the critical coupling. Let an irrational $\alpha$ be such that $|\alpha-p_n/q_n|<c q_n^{-\varkappa}$, where $p_n/q_n$, $n=1,2,\dots$ are the convergents to $\alpha$, and $c$, $\varkappa$ are positive absolute constants, $\varkappa<56$. Assuming certain conditions on the parity of the coefficients of the continued fraction of $\alpha$, we show that the central gaps of $H_{p_n/q_n}$, $n=1,2,\dots$, are inherited as spectral gaps of $H_\alpha$ of length at least $c'q_n^{-\varkappa/2}$, $c'>0$.Comment: 22 page

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

Hankel determinant and orthogonal polynomials for the Gaussian weight with a jump

We obtain asymptotics in n for the n-dimensional Hankel determinant whose symbol is the Gaussian multiplied by a step-like function. We use Riemann-Hilbert analysis of the related system of orthogonal polynomials to obtain our results.Comment: 34 pages, 7 figure

Large gap asymptotics for random matrices

We outline an approach recently used to prove formulae for the multiplicative constants in the asymptotics for the sine-kernel and Airy-kernel determinants appearing in random matrix theory and related areas.Comment: 7 page

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