Holomorphic bundles and Scalar Difference Operators: One-Point Constructions

Abstract

Commutative rings of one-dimensional difference operators of rank l>1 and their deformations are effectively constructed. Our analytical constructions are based on the so-called ''Tyurin parameters'' for the stable framed holomorphic vector bundles over algebraic curves of the genus equal to g and Chern number equal to lg. These parameters were heavily used by the present authors already in 1978-80 for the differential operators. Their deformations in the discrete case are governed by the 2D Toda Lattice hierarhy instead of KP. New integrable systems appear here in the case l=2,g=1. The theory of higher rank difference operators is much more rich than the rank one case where only 2-point constructions on the spectral curve were used in the previous literature (i.e. number of 'infinite points'' is equal to 2). One-point constructions appear in this problem for every even rank l=2k. Only in this case commutative rings depend on the functional parameters. Two-point constructions will be studied in the next work: even for higher rank l>1 this case can be solved in Theta-functions. It is not so for one-point constructions with rank l>1.