Spectral Curve of the Halphen Operator

TL;DR

This paper investigates the spectral properties of the Halphen operator, a third-order differential operator involving the Weierstrass -function, and determines the spectral curve for specific commuting operators in the equianharmonic case.

Contribution

It derives the spectral curve for the pair of commuting operators involving the Halphen operator in the equianharmonic case.

Findings

Spectral curve explicitly computed for the pair of operators.

Identification of conditions for commutation with other differential operators.

Analysis specific to the equianharmonic case where g_2=0.

Abstract

The Halphen operator is a third-order operator of the form $$ L_3=\partial_x^3-g(g+2)\wp(x)\partial_x-\frac{1}{2}g(g+2)\wp'(x), $$ where $g\ne 2\,\mbox{mod(3)}$, the Weierstrass $\wp$-function satisfies the equation $$ (\wp'(x))^2=4\wp^3(x)-g_2\wp(x)-g_3. $$ In the equianharmonic case, i.e., $g_2=0$ the Halphen operator commutes with some ordinary differential operator $L_n$ of order $n\ne 0\,\mbox{mod(3)}.$ In this paper we find the spectral curve of the pair $L_3,L_n$.