×

zbMATH — the first resource for mathematics

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
PDF BibTeX Cite
Full Text: DOI