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: ${\alpha }_j(t)=(1-|{\alpha }_j(t){|}^2)({\alpha }_{j+1}(t)-{\alpha }_{j-1}(t))$ for ${\alpha }_{-1}(t)=1$ boundary conditions and $n=0,1,2,...$, with initial condition ${\alpha }_j(0)\in (-1,1)$. We also provide examples with ${\alpha }_j(0)\in \mathbb{D}$ for which ${\alpha }_0(t)$ does not have a limit. The proofs depend on the solution via a direct/inverse spectral transform.

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