Spectral estimates for periodic Jacobi matrices

Abstract

We obtain bounds for the spectrum and for the total width of the spectral gaps for Jacobi matrices on ${\ell }^2(\mathbb{Z})$ 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 $H$.

18 pages, 5 figures, presentation improved, to appear in Commun. Math. Phys

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