Factoring Third Order Ordinary Differential Operators over Spectral Curves

Abstract

We consider the classical factorization problem of a third order ordinary differential operator L − λ, for a spectral parameter λ. It is assumed that L is an algebro-geometric operator, that it has a nontrivial centralizer, which can be seen as the affine ring of curve, the famous spectral curve Γ. In this work we explicitly describe the ring structure of the centralizer of L and, as a consequence, we prove that Γ is a space curve. In this context, the first computed example of a non-planar spectral curve arises, for an operator of this type. Based on the structure of the centralizer, we give a symbolic algorithm, using differential subresultants, to factor L − λ0 for all but a finite number of points P = (λ0, µ0, γ0) of the spectral curve .