论文标题
Darboux转换的广义方法
A Generalized Method for the Darboux Transformation
论文作者
论文摘要
提出了一种方法，以获取包含光谱参数的普通微分方程的线性系统的电势和相关波函数的变化，当该线性系统受到扰动并从光谱中添加或从光谱中添加有限数量的离散特征值时。通过引入某些基本线性积分方程，为相应的不受干扰和扰动的线性系统引入某些基本线性积分方程，从而得出了一些明确的公式。这种广义方法以统一的方式适用于广泛的线性系统。这与Darboux转换的标准方法相反，Darboux转换特定于其应用的特定线性系统。在某些特殊情况下，这种广义方法与Darboux转换的标准方法之间提供了比较。特别是，当添加一个绑定状态到离散频谱中时，为全线Schrödinger方程提供了一些Darboux转换公式，其中这些公式类似于半线Schrödinger方程的Darboux转换公式。提出的理论用一些明确的例子进行了说明。
A method is presented to obtain the change in the potential and in the relevant wavefunction of a linear system of ordinary differential equations containing a spectral parameter, when that linear system is perturbed and a finite number of discrete eigenvalues are added to or removed from the spectrum. Some explicit formulas are derived for those changes by introducing certain fundamental linear integral equations for the corresponding unperturbed and perturbed linear systems. This generalized method is applicable in a unified manner on a wide class of linear systems. This is in contrast to the standard method for a Darboux transformation, which is specific to the particular linear system on which it applies. A comparison is provided in some special cases between this generalized method and the standard method for the Darboux transformation. In particular, when a bound state is added to the discrete spectrum, some Darboux transformation formulas are presented for the full-line Schrödinger equation, where those formulas resemble the Darboux transformation formulas for the half-line Schrödinger equation. The theory presented is illustrated with some explicit examples.
