Barry Simon (California Institute of Technology)

Spectral Theory for Orthogonal Polynomials with Perturbed Periodic Recursion Coefficients

I will describe joint work with Damanik and Killip that proves analogs of the Denissov-Rakmanov theorem, Szego's theorem and the Killip-Simon theorem for perturbations about OPRL and OPUC with periodic recursion coefficients. In these results, approach to a single "free" limit is replaced by approach to an isospectral torus.
A key ingredient is to relate this approach to the approach of an associated matrix valued polynomial family to the free case. One looks at the descriminant of the periodic problem as a polynomial and proves that this polynomial evaluated on a two sided Jacobi matrix is the free matrix two sided matrix if and only if the two sided matrix lies in the isospectral torus.
These ideas reduce the periodic results to matrix results. The matrix Killip-Simon and Szego theorems are proven viq the non-local sum rules developed for scalar OPRL by Killip-Simon, Simon-Zlatos and Simon. The matrix Denissov-Rakhmanov theorem depends on results of van Assche and Marcellan.