Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2001-12-04
Theor.Math.Phys. 134 (2003) 27-38; Teor.Mat.Fiz. 134 (2003) 32-45
Nonlinear Sciences
Exactly Solvable and Integrable Systems
Write-up based on lecture by J. Harnad given at NEEDS 2001 Euroconference Isaac Newton Institute for Mathematical Sciences, Ca
Scientific paper
The two matrix model is considered, with measure given by the exponential of a sum of polynomials in two different variables. It is shown how to derive a sequence of pairs of ``dual'' finite size systems of ODEs for the corresponding biorthonormal polynomials. An inverse theorem is proved showing how to reconstruct such measures from pairs of semi-infinite finite band matrices defining the recursion relations and satisfying the string equation. A proof is given in the $N\to \infty$ limit that the dual systems obtained share the same spectral curve.
Bertola Marco
Eynard Bertrand
Harnad John
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