Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 104, 22 pages      arXiv:2303.05602      https://doi.org/10.3842/SIGMA.2023.104
Contribution to the Special Issue on Evolution Equations, Exactly Solvable Models and Random Matrices in honor of Alexander Its' 70th birthday

Szegő Kernel and Symplectic Aspects of Spectral Transform for Extended Spaces of Rational Matrices

Marco Bertola, Dmitry Korotkin and Ramtin Sasani
Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve W., Montréal, H3G 1M8 Québec, Canada

Received March 14, 2023, in final form December 02, 2023; Published online December 22, 2023

Abstract
We revisit the symplectic aspects of the spectral transform for matrix-valued rational functions with simple poles. We construct eigenvectors of such matrices in terms of the Szegő kernel on the spectral curve. Using variational formulas for the Szegő kernel we construct a new system of action-angle variables for the canonical symplectic form on the space of such functions. Comparison with previously known action-angle variables shows that the vector of Riemann constants is the gradient of some function on the moduli space of spectral curves; this function is found in the case of matrix dimension 2, when the spectral curve is hyperelliptic.

Key words: spectral transform; Szegő kernel; variational formulas.

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