Commuting Differential Operators of Rank 3 Associated to a Curve of Genus 2

In this paper, we construct some examples of commuting differential operators $L_1$ and $L_2$ with rational coefficients of rank 3 corresponding to a curve of genus 2.

💡 Research Summary

The paper addresses the long‑standing problem of constructing explicit examples of commuting ordinary differential operators of higher rank whose spectral curves have non‑trivial genus. While the theory of rank‑one and rank‑two commuting operators is well developed, concrete rank‑three cases, especially those associated with genus‑two curves, have been scarce. The authors fill this gap by presenting a systematic construction of a pair of differential operators (L_{1}) and (L_{2}) with rational coefficients, rank three, and a spectral curve of genus two.

The authors begin by recalling the Krichever‑Novikov (KN) framework: given an algebraic curve (\Gamma) and a divisor (D) of degree equal to the rank (r), one can build a Baker‑Akhiezer function (\psi(x,P)) on (\Gamma) that is meromorphic in the spectral parameter (P) and satisfies a set of linear differential equations in the spatial variable (x). The eigenvalues of the commuting operators appear as meromorphic functions (\lambda(P)) and (\mu(P)) on (\Gamma). The rank of the operator pair equals the number of linearly independent Baker‑Akhiezer functions, i.e., the dimension of the space of common eigenfunctions.

For the concrete construction, the authors choose a hyperelliptic curve of the form
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