Exactly solvable potentials with finitely many discrete eigenvalues of arbitrary choice
Exactly solvable potentials with finitely many discrete eigenvalues of arbitrary choice
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DOI：
10.1063/1.4880200
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发表时间：
2014-02
影响因子：
1.3
通讯作者：
R. Sasaki
中科院分区：
文献类型：
--
作者：
R. Sasaki
我们讨论了具有有限多个任意选择的离散本征值的精确可解势的可能变形问题。正如凯和摩西在1956年证明的那样，一维量子力学中的无反射势是完全可解的。通过附加的时间依赖关系，这些势被确定为KdV(Korteweg De Vries)系统的孤子解。N孤子势具有时间t和2N个正参数，k1<；⋯<；kN和{cj}，j=1，…，N，对应于N个离散本征值{−kj2}。特征函数是用行列式之比表示的初等函数。基于本征函数删除的Darboux-Crum-Krein-Adler变换或亚伯拉罕-摩西变换产生具有修正参数{cj‘}的较低的孤子数势。我们探索了孤子势的本征函数所满足的各种恒等式，这些恒等式反映了可分离(简并)核的Gel‘fand-Levitan-Marchenko方程的唯一性定理。
We address the problem of possible deformations of exactly solvable potentials having finitely many discrete eigenvalues of arbitrary choice. As Kay and Moses showed in 1956, reflectionless potentials in one dimensional quantum mechanics are exactly solvable. With an additional time dependence these potentials are identified as the soliton solutions of the Korteweg de Vries (KdV) hierarchy. An N-soliton potential has the time t and 2N positive parameters, k1 < ⋯ < kN and {cj}, j = 1, …, N, corresponding to N discrete eigenvalues {−kj2}. The eigenfunctions are elementary functions expressed by the ratio of determinants. The Darboux-Crum-Krein-Adler transformations or the Abraham-Moses transformations based on eigenfunction deletions produce lower soliton number potentials with modified parameters {cj′}. We explore various identities satisfied by the eigenfunctions of the soliton potentials, which reflect the uniqueness theorem of Gel'fand-Levitan-Marchenko equations for separable (degenerate) kernels.