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February  2007, 1(1): 189-215. doi: 10.3934/ipi.2007.1.189

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, United States

Received  September 2006 Revised  October 2006 Published  January 2007

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.
Keywords: Toda flows., orthogonal polynomials, Schur flows.
Mathematics Subject Classification: 05E35, 33D45, 37K1.
Citation: Barry Simon. Zeros of OPUC and long time asymptotics of Schur and related flows. Inverse Problems & Imaging, 2007, 1 (1) : 189-215. doi: 10.3934/ipi.2007.1.189
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