arXiv 1402.54742014-02-22math-phhep-thmath.CAmath.MP

具有任意选择的多个离散特征值的精确可解位势

Exactly solvable potentials with finitely many discrete eigenvalues of arbitrary choice

Ryu Sasaki

备注：: LaTeX 17 pages, no figure
DOI：: 10.1063/1.4880200

摘要：

我们解决的问题，可能的变形的精确可解的潜力有许多离散的特征值的任意选择。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。与一个额外的时间依赖性，这些潜力被确定为孤子解的KdV系列。一个$N$ -孤子势具有时间$t$和$2N$正参数$k_1<...<k_N$和$c_j$，$j=1,...,N$，对应于$N$离散本征值$-k_j^2$ .本征函数是由行列式的比值表示的初等函数。基于本征函数删除的Darboux-Crum-Krein-Adler变换或Abraham-Moses变换在修正参数$c'_j$的情况下产生较低的孤子数势。我们探讨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（退化）核的Gel'fand-Levitan-Marchenko方程的唯一性定理.

英文摘要：

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 KdV hierarchy. An $N$-soliton potential has the time $t$ and $2N$ positive parameters, $k_1<...<k_N$ and $c_j$, $j=1,...,N$, corresponding to $N$ discrete eigenvalues $-k_j^2$. The eigenfunctions are elementary functions expressed by the ratio of determinants. The Darboux-Crum-Krein-Adler transformation or the Abraham-Moses transformations based on eigenfunctions deletions produce lower soliton number potentials with modified parameters $c'_j$. 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.

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英文摘要：

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