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.

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 |