Factorization of KdV Schrödinger operators using differential subresultants.

We address the classical factorization problem of a one dimensional Schrödinger operator −∂2+u−λ, for a stationary potential u of the KdV hierarchy but, in this occasion, a “parameter” λ is considered. Inspired by the more effective approach of Gesztesy and Holden to the “direct” spectral problem in [14], we give a symbolic algorithm by means of differential elimination tools to achieve the aimed factorization, that in addition allows one parameter form factorizations by means of suitable global parametrizations of the spectral curve. Differential resultants are used for computing spectral curves, and differential subresultants to obtain the first order common factor. To make our method fully effective, we design a symbolic algorithm to compute the integration constants of the KdV hierarchy, in the case of KdV potentials that become rational under a Hamiltonian change of variable. Explicit computations are carried for Schrödinger operators with solitonic potentials.