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

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 $α_j(0)\in (-1,1)$. We also provide examples with $α_j(0)\in\mathbb{D}$ for which $α_0(t)$ does not have a limit. The proofs depend on the solution via a direct/inverse spectral transform.