Zeros of OPUC and Long Time Asymptotics of Schur and Related Flows

2006/10/31 by Barry Simon, Simon, Barry
Mathematics · Physics and Astronomy · #05E25 #37K10 #39A11 #FOS: Mathematics #Random Matrices and Applications #Spectral Theory (math.SP) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.SP #msc:05E25 #msc:37K10 #msc:39A11

paper · pdf · doi:10.48550/arxiv.math/0610987

arxiv created 2006/10/31 · openalex publication_date 2006/10/31 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We provide a complete analysis of the asymptotics for the semi-infinite Schur flow: α_j(t)=(1- |α_j(t)|²) (α_j+1(t)-α_j-1(t)) for α_-1(t)= 1 boundary conditions and n=0,1,2,..., with initial condition α_j(0)∈ (-1,1). We also provide examples with α_j(0)∈\mathbbD for which α₀(t) does not have a limit. The proofs depend on the solution via a direct/inverse spectral transform.

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