Zeros of OPUC and long time asymptotics of Schur and related flows

Barry Simon

doi:10.3934/ipi.2007.1.189

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2007, Volume 1, Issue 1: 189-215. Doi: 10.3934/ipi.2007.1.189

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Zeros of OPUC and long time asymptotics of Schur and related flows

Barry Simon^1,

1.
Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125

Received: August 31, 2006

Revised: September 30, 2006

Published: January 2007

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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 $\alpha_j(0)\in\bbD$ 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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Mathematics Subject Classification: 05E35, 33D45, 37K10.

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