Spectral estimates for periodic Jacobi matrices

摘要

We obtain bounds for the spectrum and for the total width of the spectral gaps, for Jacobi matrices on l(2)(Z) of the form (Hpsi)(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 abased 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. [References: 20]