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Commuting difference operators with polynomial coefficients. (English. Russian original) Zbl 1139.39037
Russ. Math. Surv. 62, No. 4, 819-820 (2007); translation from Usp. Mat. Nauk 62, No. 4, 169-170 (2007).
From the introduction: In the framework of the one-point construction, we find here the commuting difference operators of rank 2 that correspond to a curve of genus 1 and have coefficients that are polynomials in a discrete variable.
The joint eigenfunctions of two commuting difference operators
\[ L_1= \sum_{N_-}^{N_+} u_i(n)T^i, \qquad L_2= \sum_{M_-}^{M_+} v_i(n)T^i, \]
where \(T\) is the shift operator with respect to the discrete variable \(n\in\mathbb Z\), are parametrized by the spectral curve \(\Gamma\) given in \(\mathbb C^2\) by some polynomial \(Q(\lambda,\mu)\):
\[ L_1\psi(n,P)= \lambda\psi(n,P), \quad L_2\psi(n,P)= \mu\psi(n,P), \quad P=(\lambda,\mu)\in \Gamma. \]
By the rank \(l\) of the pair of operators \(L_1, L_2\) we mean the dimension of the space of joint eigenfunctions at the point \(P\in\Gamma\) in general position. For \(l=1\) the function \(\psi(n,P)\) can be expressed in terms of the theta function of the curve \(\Gamma\), and one can readily recover the coefficients of the operators. For \(l>1\) the problem of finding the function \(\psi(n,P)\) reduces to the solution of a Riemann problem, and it has not been possible to find this function explicitly.
39A70 Difference operators
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
14K05 Algebraic theory of abelian varieties
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