Remarks on periodic Jacobi matrices on trees

Seminar on Jacobi Matrices

Marchenko-Ostrovski mappings for periodic Jacobi matrices

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.

Remarks on Contact and Jacobi Geometry

We present an approach to Jacobi and contact geometry that makes many facts, presented in the literature in an overcomplicated way, much more natural and clear. The key concepts are Kirillov manifolds and linear Kirillov structures, i.e., homogeneous Poisson manifolds and, respectively, homogeneous linear Poisson manifolds. The difference with the existing literature is that the homogeneity of ...

Some Remarks on Fibonacci Matrices

In [1], Dazheng studies Fibonacci matrices, namely matrices M such that every entry of every positive power of M is either 0 or plus or minus a Fibonacci number. He gives 40 such four-byfour matrices. In the following, we give an interpretation of these matrices, from which we give simpler proofs of several of his theorems. We also determine all two-by-two Fibonacci matrices. Let £ = e be a pri...

Limit Periodic Jacobi Matrices with a Singular Continuous Spectrum and the Renormalization of Periodic Matrices

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 dim...