Lyapunov functions of periodic matrix-valued Jacobi operators (original) (raw)

Marchenko-Ostrovski mappings for periodic Jacobi matrices

Russian Journal of Mathematical Physics, 2007

We consider the 1D periodic Jacobi matrices. The spectrum of this operator is purely absolutely continuous and consists of intervals separated by gaps. We solve the inverse problem (including characterization) in terms of vertical slits on the quasimomentum domain. Furthermore, we obtain a priori two-sided estimates for vertical slits in terms of Jacoby matrices.

The Spectrum of Periodic Jacobi Matrices with Slowly Oscillating Diagonal Terms

We study the spectrum of periodic Jacobi matrices. We concentrate on the case of slowly oscillating diagonal terms and study the behaviour of the zeros of the associated orthogonal polynomials in the spectral gap. We find precise estimates for the distance from single eigenvalues of truncated matrices in the spectral gap to the diagonal entries of the matrix. We include a brief numerical example to show the exactness of our estimates.

Singular Continuous Spectrum for a Class of Almost Periodic Jacobi MATRICES1

AMERICAN MATHEMATICAL SOCIETY, 1982

We consider the operator// on l2(Z) depending upon three parameters, X, a, 0, ... (1) [#(X, a, 0)u] (n) = u(n + 1) + u(n - 1) + X cos(2iran + 6)u(n). ... In this note we will sketch the proof of the following result whose detailed proof will appear elsewhere [3]. ... THEOREM 1. Fix a, an ...

On stability in the Borg–Hochstadt theorem for periodic Jacobi matrices

Journal of Spectral Theory

A result of Borg-Hochstadt in the theory of periodic Jacobi matrices states that such a matrix has constant diagonals as long as all gaps in its spectrum are closed (have zero length). We suggest a quantitative version of this result by proving the two-sided bounds between oscillations of the matrix entries along the diagonals and the length of the maximal gap in the spectrum.