Abstract

By combining ideas of Lubinsky with some soft analysis, we prove that universality and clock behavior of zeros for orthogonal polynomials on the real line in the absolutely continuous spectral region is implied by convergence of K_n(x,x) for the diagonal CD kernel and boundedness of the analog associated to second kind polynomials. We then show that these hypotheses are always valid for ergodic Jacobi matrices with absolutely continuous spectrum and prove that the limit of K_n(x,x) is ρ_∞(x) ∕ w(x), where ρ_∞ is the density of zeros and w is the absolutely continuous weight of the spectral measure.

Keywords

orthogonal polynomials, clock behavior, almost Mathieu equation

Mathematical Subject Classification 2000

Primary: 26C10, 42C05, 47B36

Milestones

Received: 20 October 2009
Accepted: 19 November 2009
Published: 4 March 2010

Authors