# Factorization of KdV Schr\"odinger operators using differential subresultants

## TL;DR

This paper presents a symbolic algorithm leveraging differential elimination tools to factorize KdV Schr"odinger operators with stationary potentials, using differential resultants and subresultants to compute spectral curves and factors.

## Contribution

It introduces a novel symbolic method for factorizing KdV Schr"odinger operators using differential subresultants, extending previous approaches with an effective algorithm for integration constants.

## Key findings

- Successfully factorized Schr"odinger operators with solitonic potentials
- Developed an effective symbolic algorithm for the KdV hierarchy constants
- Computed spectral curves using differential resultants

## Abstract

We address the classical factorization problem of a one dimensional Schr\"odinger operator $-\partial^2+u-\lambda$, for a stationary potential $u$ of the KdV hierarchy but, in this occasion, a "parameter" $\lambda$. Inspired by the more effective approach of Gesztesy and Holden to the "direct" spectral problem, we give a symbolic algorithm by means of differential elimination tools to achieve the aimed factorization. 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 Schr\"odinger operators with solitonic potentials.