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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
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 \[ \gamma_1(n)+\dots+\gamma_g(n)-P_{-1}-\dots-P_{-n}-\gamma_1-\dots-\gamma_g-nq, \]
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.

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