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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, Nemytskiĭ, Uryson, etc.) 39A05 General theory of difference equations
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##### References:
  Krichever, I. M., No article title, Russ. Math. Surveys, 33, 255-256, (1978) · Zbl 0412.39002  Krichever, I. M.; Novikov, S. P., No article title, Russ. Math. Surveys, 58, 473-510, (2003) · Zbl 1060.37068  D. Mumford, in Proceeding of International Symposium on Algebraic Geometry, Kyoto, Japan, 1977 (Kinokuniya, Tokio, 1978), pp. 115-153.  Mauleshova, G. S.; Mironov, A. E., No article title, Russ. Math. Surveys, 70, 557-559, (2015) · Zbl 1326.47037  Krichever, I. M., No article title, Dokl. Akad. Nauk SSSR, 285, 31-36, (1985)  Krichever, I. M., No article title, Funct. Anal. Appl., 49, 175-188, (2015) · Zbl 1331.47052  Dixmier, J., No article title, Bull. Soc. Math. France, 96, 209-242, (1968)  Kanel-Belov, A. Ya.; Kontsevich, M. L., No article title, Mosc. Math. J., 7, 209-218, (2007)  A. E. Mironov and A. B. Zheglov, IMRN (2015); doi:10.1093/imrn/rnv218.
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