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One-point commuting difference operators of rank 1. (English. Russian original) Zbl 1361.13012
Dokl. Math. 93, No. 1, 62-64 (2016); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 466, No. 4, 399-401 (2016).
Consider space of functions from \({\mathbb Z}\) to \({\mathbb C}\). Let \(T\) be a shift operator, and \(L_k=\sum_{j=-K^-}^{K^+}u_j(n)T^n\) be a shift operator of order \(k=K^+ +K^-\). If two such operators \(L_k\) and \(L_m\) commute, then there exist a polynomial \(R(L_k,L_m)=0\) defining a spectral curve \(\Gamma\). If \(L_k(\psi)=z\psi\) and \(L_m(\psi)=w\psi\) then \(R(z,w)=0\), i.e. \((z,w)\in \Gamma\). The paper announces the following result. Consider a spectral data \[ S=\{\Gamma,\gamma_1,\dots,\gamma_g,k^{-1},P_n\}, \] where \(\Gamma\) is a curve of genus \(g\), \(\gamma=\gamma_1+\dots+\gamma_g\) is a special divisor on \(\Gamma\), \(q\in\Gamma\), \(t^{-1}\) is a local parameter near \(q\), \(\{P_n\in\Gamma\}\) is a set of points.
Then there exists a unique Baker-Akhiezer function \(\psi(n,P)\) such that
{\(\bullet\)}
the divisor of \(\psi\) for \(n\geq 0\) has the form \[ \gamma_1(n)+\dots+\gamma_g(n)+P_1+\dots+P_n-\gamma_1-\dots-\gamma_g-nq, \] and for \(n< 0\) has the form \[ \gamma_1(n)+\dots+\gamma_g(n)-P_{-1}-\dots-P_{-n}-\gamma_1-\dots-\gamma_g-nq, \]
{\(\bullet\)}
in a neighbourhood of \(q\) the function \(\psi\) expands as \(\psi=k^n+O(k^{n-1})\). For any meromorphic functions \(f(P)\) and \(g(P)\) with unique poles of orders \(m\) and \(s\) there exist a a commuting difference operators \(L_m\) and \(L_s\) such that \(L_m(\psi)=f(P)\psi\), \(L_s(\psi)=g(P)\psi\).

This result has different applications. In particular, for hyperelliptic and elliptic spectral curve constructions. Relations of commuting difference operators with Weil algebra automorphism and Dixmier conjecture are briefly discoursed.

MSC:
13N15 Derivations and commutative rings
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiń≠, Uryson, etc.)
39A05 General theory of difference equations
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