Lyapunov functions of periodic matrix-valued Jacobi operators (original) (raw)
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.
2005
For all hyperbolic polynomials we proved in [11] a Lipschitz estimate of Jacobi matrices built by orthogonalizing polynomials with respect to measures in the orbit of classical Perron-Frobenius-Ruelle operators associated to hyperbolic polynomial dynamics (with real Julia set). Here we prove that for all sufficiently hyperbolic polynomials this estimate becomes exponentially better when the dimension of the Jacobi matrix grows. In fact, our main result asserts that a certain natural non-linear operator on Jacobi matrices built by a hyperbolic polynomial with real Julia set is a contraction in operator norm if the polynomial is sufficiently hyperbolic. This allows us to get for such polynomials the solution of a problem of Bellissard, in other words, to prove the limit periodicity of the limit Jacobi matrix. This fact does not require the iteration of the same fixed polynomial, and therefore it gives a wide class of limit periodic Jacobi matrices with singular continuous spectrum.
Inverse spectral theory for Jacobi matrices and their almost periodicity
1994
In this paper we consider the inverse problem for bounded Jacobi matrices with nonempty absolutely continuous spectrum and as an application show the almost periodicity of some random Jacobi matrices. We do the inversion in two different ways. In the general case we use a direct method of reconstructing the Green functions. In the special case where we show the almost periodicity, we use an alternative method using the trace formula for points in the orbit of the matrices under translations. This method of reconstruction involves analyzing the Abel-Jacobi map and solving of the Jacobi inversion problem associated with an infinite genus Riemann surface constructed from the spectrum. Contents 1 Introduction 2 1.1 Ideas, strategies and limitations : : : : : : : : : : : : : : : : : 4 2 Inverse Spectral Theory 8 3 Interpolation theorem 18 4 Analysis on a Riemann surface 32 4.1 The Riemann Surface : : : : : : : : : : : : : : : : : : : : : : : 32 4.2 The Abel-Jacobi map : : : : : : : : :...
The Lyapunov function for Schrödinger operators with a periodic matrix potential
Journal of Functional Analysis, 2006
We consider the Schrödinger operator on the real line with a 2 × 2 matrix-valued 1-periodic potential. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define a Lyapunov function which is analytic on a two-sheeted Riemann surface. On each sheet, the Lyapunov function has the same properties as in the scalar case, but it has branch points, which we call resonances. We prove the existence of real as well as non-real resonances for specific potentials. We determine the asymptotics of the periodic and the anti-periodic spectrum and of the resonances at high energy. We show that there exist two type of gaps: (1) stable gaps, where the endpoints are the periodic and the anti-periodic eigenvalues, (2) unstable (resonance) gaps, where the endpoints are resonances (i.e., real branch points of the Lyapunov function). We also show that periodic and anti-periodic spectrum together determine the spectrum of the matrix Hill operator.