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.

中文翻译：

---

使用微分子结果对 KdV 薛定谔算子进行因式分解

我们解决了一个一维 Schr\"odinger 算子 $-\partial^2+u-\lambda$ 的经典分解问题，对于 KdV 层次结构的固定势 $u$，但在这种情况下，一个“参数”$ λ$。受 Gesztesy 和 Holden 对“直接”谱问题更有效的方法的启发，我们通过微分消除工具给出了一个符号算法来实现目标分解。微分结果用于计算谱曲线，并微分子结果以获得一阶公因子。为了使我们的方法完全有效，我们设计了一个符号算法来计算 KdV 层次结构的积分常数，在 KdV 势在变量的哈密顿变化下变为有理数的情况下。显式计算是为 Schr 携带\"具有孤子势的odinger算子。