Normal matrix models, dbar-problem, and orthogonal polynomials on the complex plane

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Its, Alexander R.
Takhtajan, Leon A.
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Abstract
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We introduce a dbar-formulation of the orthogonal polynomials on the complex plane, and hence of the related normal matrix model, which is expected to play the same role as the Riemann-Hilbert formalism in the theory of orthogonal polynomials on the line and for the related Hermitian model. We propose an analog of Deift-Kriecherbauer-McLaughlin-Venakides-Zhou asymptotic method for the analysis of the relevant dbar-problem, and indicate how familiar steps for the Hermitian model, e.g. the g-function ``undressing'', might look like in the case of the normal model. We use the particular model considered recently by P. Elbau and G. Felder as a case study.
Comment: 14 pages
Keywords
Mathematics - Classical Analysis and ODEs, Mathematical Physics, Nonlinear Sciences - Exactly Solvable and Integrable Systems
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