Jacobi matrices with lacunary spectrum

The absolutely continuous spectrum of Jacobi matrices

I explore some consequences of a groundbreaking result of Breimesser and Pearson on the absolutely continuous spectrum of one-dimensional Schrödinger operators. These include an Oracle Theorem that predicts the potential and rather general results on the approach to certain limit potentials. In particular, we prove a Denisov-Rakhmanov type theorem for the general finite gap case. The main theme...

Some Jacobi Matrices with Decaying Potential and Dense Point Spectrum

We discuss doubly infinite matrices of the form Mi — δi + 1 -\-δί ._ t + Ff<5i;. as operators on / 2 . We present for each 8>0, examples of potentials Vn with \Vn\ = O(n~ ί/2 + ) and where M has only point spectrum. Our discussion should be viewed as a remark on the recent work of Delyon, Kunz, and Souillard.

Two modified Jacobi methods for M-matrices

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...

The Discrete Spectrum for Complex Perturbations of Periodic Jacobi Matrices

We study spectrum inclusion regions for complex Jacobi matrices which are compact perturbations of real periodic Jacobi matrix. The condition sufficient for the lack of discrete spectrum for such matrices is given.