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Holomorphic bundles and commuting difference operators. Two-point constructions. (English. Russian original) Zbl 0978.35066
Russ. Math. Surv. 55, No. 3, 586-588 (2000); translation from Usp. Mat. Nauk 55, No. 3, 181-182 (2000).
From the introduction: In our previous paper [Russ. Math. Surv. 55, 180–181 (2000); translation from Usp. Mat. Nauk 55, 187–188 (2000; Zbl 1101.14315)] we showed that for rank $$2l\geq 2$$ a broad class of commuting difference operators can be obtained from the one-point construction. As in the continuous case, these operators depend on arbitrary functions of one variable $$n\in \mathbb Z$$.
In the present paper we obtain a description of a broad class of commuting difference operators constructed starting from two-point constructions. In contrast to one-point constructions, there are no arbitrary functions here; the coefficients of the operators can be calculated by means of the Riemann theta function. As in the rank 1 case, these operators lead to solutions of the equations of the 2D Toda lattice and the whole hierarchy connected with them.

MSC:
 37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions 14H70 Relationships between algebraic curves and integrable systems 39A70 Difference operators 47B39 Linear difference operators
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