  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.
