Exactly solvable potentials with finitely many discrete eigenvalues of arbitrary choice

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.