On commutative subalgebras of the Weyl algebra that are related to commuting operators of arbitrary rank and genus

We construct examples of commuting ordinary scalar differential operators with polynomial coefficients that are related to a spectral curve of an arbitrary genus g and to an arbitrary even rank r = 2k, and also to an arbitrary rank of the form r = 3k, of the vector bundle of common eigenfunctions of the commuting operators over the spectral curve.

💡 Research Summary

The paper addresses the long‑standing problem of constructing explicit pairs of commuting ordinary scalar differential operators with polynomial coefficients within the first Weyl algebra (W_1(\mathbb{C})). While the existence of such pairs for a given algebraic spectral curve and a vector bundle of eigenfunctions is guaranteed by the Krichever‑Novikov (KN) theory, concrete examples for arbitrary genus (g) and high rank (r) have been scarce. The authors fill this gap by presenting a systematic construction that works for any genus (g\ge 1) and for ranks of the form (r=2k) (even) or (r=3k) (a multiple of three), where (k) is an arbitrary positive integer.

The construction begins with an algebraic curve (\Gamma) of genus (g), defined by a hyperelliptic equation (y^2=P_{2g+1}(\lambda)) or (y^2=P_{2g+2}(\lambda)) with a polynomial (P) of the appropriate degree. Over this curve the authors consider a holomorphic vector bundle (E) of rank (r). The common eigenfunction (\psi(x,\lambda)) of the commuting operators satisfies the Baker‑Akhiezer type relations \