An approach to universality using Weyl mfunctions
Abstract
We describe an approach to universality limits for orthogonal polynomials on the real line which is completely local and uses only the boundary behavior of the Weyl mfunction at the point. We show that bulk universality of the ChristoffelDarboux kernel holds for any point where the imaginary part of the mfunction has a positive finite nontangential limit. This approach is based on studying a matrix version of the ChristoffelDarboux kernel and the realization that bulk universality for this kernel at a point is equivalent to the fact that the corresponding mfunction has normal limits at the same point. Our approach automatically applies to other selfadjoint systems with $2\times 2$ transfer matrices such as continuum Schrödinger and Dirac operators. We also obtain analogous results for orthogonal polynomials on the unit circle.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 DOI:
 10.48550/arXiv.2108.01629
 arXiv:
 arXiv:2108.01629
 Bibcode:
 2021arXiv210801629E
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematical Physics;
 Mathematics  Spectral Theory;
 42C05 (Primary);
 47B36;
 34L40;
 46E22;
 47B32 (Secondary)
 EPrint:
 29 pages. v2 also contains results for OPUC