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BULK UNIVERSALITY AND CLOCK SPACING OF ZEROS FOR ERGODIC JACOBI MATRICES
 

Summary: BULK UNIVERSALITY AND CLOCK SPACING
OF ZEROS FOR ERGODIC JACOBI MATRICES
WITH A.C. SPECTRUM
ARTUR AVILA1
, YORAM LAST2,4
, AND BARRY SIMON3,4
Abstract. By combining some ideas of Lubinsky with some soft
analysis, we prove that universality and clock behavior of zeros
for OPRL in the a.c. spectral region is implied by convergence
of 1
n Kn(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
a.c. spectrum and prove that the limit of 1
n Kn(x, x) is (x)/w(x)
where is the density of zeros and w is the a.c. weight of the
spectral measure.
1. Introduction
Given a finite measure, dµ, of compact and not finite support on R,
one defines the orthonormal polynomials, pn(x) (or pn(x, dµ) if the µ-

  

Source: Avila, Artur - Instituto Nacional de Matemática Pura e Aplicada (IMPA)
Mathematical Physics Preprint Archive

 

Collections: Mathematics