Holomorphic bundles and Scalar Difference Operators: OnePoint Constructions
Abstract
Commutative rings of onedimensional difference operators of rank l>1 and their deformations are effectively constructed. Our analytical constructions are based on the socalled ''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 197880 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 2point constructions on the spectral curve were used in the previous literature (i.e. number of 'infinite points'' is equal to 2). Onepoint constructions appear in this problem for every even rank l=2k. Only in this case commutative rings depend on the functional parameters. Twopoint constructions will be studied in the next work: even for higher rank l>1 this case can be solved in Thetafunctions. It is not so for onepoint constructions with rank l>1.
 Publication:

arXiv eprints
 Pub Date:
 April 2000
 arXiv:
 arXiv:mathph/0004008
 Bibcode:
 2000math.ph...4008K
 Keywords:

 Mathematical Physics;
 Mathematics  Mathematical Physics
 EPrint:
 Latex2e file, 4 pages