Tyurin parameters of commuting pairs and infinite dimensional Grassmann manifold

Abstract

Commuting pairs of ordinary differential operators are classified by a set of algebro-geometric data called ``algebraic spectral data''. These data consist of an algebraic curve (``spectral curve'') $\Gamma$ with a marked point ${\gamma }_{\infty }$, a holomorphic vector bundle $E$ on $\Gamma$ and some additional data related to the local structure of $\Gamma$ and $E$ in a neighborhood of ${\gamma }_{\infty }$. If the rank $r$ of $E$ is greater than 1, one can use the so called ``Tyurin parameters'' in place of $E$ itself. The Tyurin parameters specify the pole structure of a basis of joint eigenfunctions of the commuting pair. These data can be translated to the language of an infinite dimensional Grassmann manifold. This leads to a dynamical system of the standard exponential flows on the Grassmann manifold, in which the role of Tyurin parameters and some other parameters is made clear.

latex2e, use amssymb and amsmath packages, contribution to proceedings of RIMS workshop "Elliptic Integrable Systems" (RIMS, 2004)

References in corpus (1)

• Tyurin parameters and elliptic analogue of nonlinear Schrödinger hierarchy