Spectral estimates for periodic Jacobi matrices

Details Spectral estimates for periodic Jacobi matrices Spectral estimates for periodic Jacobi matrices

2002-05-30

arxiv.org/abs/math/0205319v3

Commun. Math. Phys. 234, 517-532 (2003)

Mathematics

Spectral Theory

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

Scientific paper

10.1007/s00220-002-0768-2

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 $H$.

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