Discretization of Commuting Ordinary Differential Operators of Rank 2 in the Case of Elliptic Spectral Curves
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scretization of Commuting Ordinary Differential Operators of Rank 2 in the Case of Elliptic Spectral Curves Gulnara S. Mauleshova a,b and Andrey E. Mironov a,b Received December 1, 2019; revised December 1, 2019; accepted May 21, 2020
To Valery Vasil’evich Kozlov on the occasion of his 70th birthday Abstract—We study one-point commuting difference operators of rank 2 and establish a relationship between these operators and commuting differential operators of rank 2 in the case of elliptic spectral curves. DOI: 10.1134/S0081543820050168
1. INTRODUCTION AND MAIN RESULTS In this paper we continue the study of one-point commuting difference operators of rank 2 (Krichever–Novikov operators; see [11]), which began in [14, 18]. Using the results obtained in this paper, we construct a discretization of commuting ordinary differential operators of rank 2 in the case of elliptic spectral curves. This discretization has the following remarkable properties. When the shift parameter tends to zero, the difference operators turn into differential ones, while the spectral curve does not depend on the shift parameter and coincides with the spectral curve of the differential operators. Denote by Lp and Lm difference operators of orders p and m of the form N+
Lp =
i
ui (n)T ,
Lm =
i=−N−
M+
vj (n)T j ,
j=−M−
where p = N+ + N− , m = M+ + M− , N± , M± ∈ N, and T is the shift operator, T f (n) = f (n + 1). The operators Lp and Lm act on a set of functions {f : Z → C}. If Lp and Lm commute, then there exists a nonzero polynomial F (z, w) such that F (Lp , Lm ) = 0 (see [8]). This polynomial defines a spectral curve Γ = {(z, w) ∈ C2 : F (z, w) = 0}. The rank l of the pair Lp , Lm is the number l = dim{ψ : Lp ψ = zψ, Lm ψ = wψ, (z, w) ∈ Γ}; here it is assumed that the point (z, w) is in general position on Γ. Any maximal commutative ring of difference operators containing Lp and Lm is isomorphic to a ring of meromorphic functions on some algebraic curve with poles at points q1 , . . . , qs (see [11]). Two-point (i.e., s = 2) rank 1 operators were studied in [7, 9, 19, 23]. In this paper we consider one-point (i.e., s = 1) operators of rank 2 of the form L4 =
2
ui (n)T i ,
i=−2
L4g+2 =
2g+1
vi (n)T i ,
u2 = v2g+1 = 1,
(1.1)
i=−(2g+1)
a Novosibirsk State University, ul. Pirogova 1, Novosibirsk, 630090 Russia. b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, pr. Akad. Koptyuga 4,
Novosibirsk, 630090 Russia. E-mail addresses: [email protected] (G. S. Mauleshova), [email protected] (A. E. Mironov).
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DISCRETIZATION OF COMMUTING DIFFERENTIAL OPERATORS
203
with a hyperelliptic spectral curve Γ of genus g given by the equation w2 = Fg (z) = z 2g+1 + c2g z 2g + c2g−1 z 2g−1 + . . . + c0 ,
(1.2)
with the common eigenfunctions satisfying the equations L4 ψ = zψ,
L4g+2 ψ = wψ,
ψ = ψ(n, P ),
P = (z, w) ∈ Γ.
(1.3)
The common eigenfunction ψ(n, P ) of the operators L4 and L4g+2 (Baker–Akhiezer function) also satisfies the equation ψ(n + 1, P ) = χ1 (n, P )ψ(n − 1, P ) + χ2 (n, P )ψ
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