Spectral estimates for periodic Jacobi matrices

E. Korotyaev; I. V. Krasovsky

doi:10.1007/s00220-002-0768-2

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arXiv · math/0205319

Spectral estimates for periodic Jacobi matrices

E. Korotyaev·I. V. Krasovsky

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Abstract

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 $(Hψ)_n= a_{n-1}ψ_{n-1}+b_nψ_n+a_nψ_{n+1}$, 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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BibTeX RIS

E. Korotyaev, I. V. Krasovsky. 2003-10-13. Spectral estimates for periodic Jacobi matrices. https://doi.org/10.1007/s00220-002-0768-2

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