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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.
##### MSC:
 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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