Reconstruction of confining or attractive potentials of one-dimensional Schrödinger equation

The dependence of confining or attractive potentials on the characteristics of the discrete spectrum of the one-dimensional Schrödinger equation is investigated. Asymptotic expressions are obtained for the regular solutions and the corrections to the potential in different cases of variation of the spectral characteristics. The connection between one-parameter transformations of the normalization constants and the nonlinear evolution equations for the recovered potential is considered.