Commuting differential operators of rank 2 and arbitrary genusgwith polynomial coefficients

Abstract

If two differential operators Ln = n i=0 ui(x)∂ i x and Lm = m i=0 vi(x)∂ i x commute, then there is a non-zero polynomial R(z, w) such that R(Ln, Lm) = 0 (see [1]). The curve Γ defined by R(z, w) = 0 is called the spectral curve of the pair. If Lnψ = zψ and Lmψ = wψ, then (z, w) ∈ Γ. The dimension of the space of common eigenfunctions ψ is the same for almost all (z, w) ∈ Γ. The dimension of the space of common eigenfunctions of two commuting differential operators is called the rank of the pair. The rank is a common divisor of m and n. The coefficients of two commuting operators of rank 1 can be expressed explicitly in terms of Riemann theta-functions [2]. The case when the rank is greater than 1 is much more difficult. The first examples of commuting differential operators of non-trivial rank 2 and non-trivial genus g = 1 were constructed by Dixmier in [3] for the non-singular elliptic spectral curve w 2 = z 3 − α with an arbitrary non-zero constant α:where L, M is the commuting pair of Dixmier operators of rank 2 and genus 1. A general classification of pairs of commuting ordinary differential operators of rank greater than 1 was obtained by Krichever [4]. The general form of pairs of commuting operators of rank 2 for an arbitrary elliptic spectral curve was found by Krichever and Novikov [5]. The general form of pairs of operators of rank 3 for an arbitrary elliptic spectral curve (the general form of commuting operators of rank 3 and genus 1 is parametrized by two arbitrary functions) was found by Mokhov [6], [7]. Moreover, examples of commuting differential operators of genus 1 with polynomial coefficients have been constructed for any rank. However, even in those cases for which explicit formulae were obtained, the problem of determining commuting operators with polynomial coefficients is non-trivial and has not yet been completely solved. The problem of a complete description of commuting differential operators with polynomial coefficients was posed and discussed by Mokhov in [8]. In [9] examples were constructed of pairs of commuting operators L and M of rank 2 and arbitrary genus g:where the Ai are arbitrary constants, A3 ̸ = 0, and the ai are some constants. Examples of commuting differential operators of arbitrary genus and arbitrary rank with polynomial coefficients were constructed in [10]. In this paper we find new examples of pairs of commuting operators of rank 2. We have proved the following three theorems.

Commuting Differential Operators of Rank 2 with Rational Coefficients

“…Common eigenfunction of operators L 1 , M 1 and L 2 , M 2 in some special cases were found in [18]. Moreover, the following theorems are proved in [15], [16].…”

The AKNS Hierarchy and Finite-Gap Schrödinger Potentials

“…where a i is a pole of functions p(x) and q(x). Substituting y in (19) and considering the coefficient of (x − a i ) σ−1 we get…”

“…Mironov in [7] found new methods of constructing new scalar commuting operators of rank 2. Using Mironov's method many examples of scalar commuting operators of rank 2 were found (see [11], [19] [21], [14], [18], [20]). A general classification of commuting matrix differential operators was obtained by Grinevich [22].…”

Explicit Characterization of Some Commuting Differential Operators of Rank 2

“…The spectral curves of these operators have the form w 2 = z 2m+1 + a 2m z 2m + ... + a 1 z + a 0 . Moreover, the following theorems are proved in [12], [13].…”

“…Operators L 1 and L 2 commute with operators of order 4g+2 (see [12], [13]). The spectral curves of these operators have the form w 2 = z 2m+1 + a 2m z 2m + ... + a 1 z + a 0 .…”