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Spectral Curve of the Halphen Operator

Part of: Ordinary differential operators

Published online by Cambridge University Press:  14 November 2016

Andrey E. Mironov  and
Dafeng Zuo
Andrey E. Mironov
Affiliation:
Sobolev Institute of Mathematics, 4 Acad. Koptyug avenue, 630090 Novosibirsk, Russia (mironov@math.nsc.ru)
Dafeng Zuo
Affiliation:
Wu Wen-Tsun Key Laboratory of Mathematics, Chinese Academy of Sciences, School of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, People's Republic of China (dfzuo@ustc.edu.cn)
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Abstract

The Halphen operator is a third-order operator of the form

where g ≠ 2 mod(3), where the Weierstrass ℘-function satisfies the equation

In the equianharmonic case, i.e. g2 = 0, the Halphen operator commutes with some ordinary differential operator Ln of order n ≠ 0 mod(3). In this paper we find the spectral curve of the pair L3, Ln.

Keywords

commuting ordinary differential operatorsspectral curveHalphen operator

MSC classification

Secondary: 34L05: General spectral theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 2 , May 2017 , pp. 451 - 460
Copyright
Copyright © Edinburgh Mathematical Society 2017 

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