Dependence of potential on characteristics of discrete spectrum in quantum mechanics

Using a general solution of the inverse problem for tbe reconstruction of a confining potential $(\tilde{V}(r) \to \infty, r \to \infty$) from a reference potential $V(r)$, energy levels and normalizing constants, we analyze the corrections to the potential $\Delta V(r)$ and to the regular solutions of the radial Schrodinger equation when a finite number of the spectral characteristics are changed. For $r \to 0$ and any arbitrary changes of the spectrum $\Delta V(r) \sim r^{2l+1}$ ($l$ is the orbital angular momentum). For $r \to \infty$, the asymptotics of $\Delta V(r)$ are classified according to three different types of change of the spectral characteristics. It is shown that when additional energy levels are included or some energy levels are omitted, there is a one-to-one relationship between the asymptotic behavior of $\Delta V(r)$ and the period of radial oscillations of the classical motion in the field $V(r)$. All results can be carried through to the case of long-range attractive potentials ($\lim_{r\to\infty} = 0$, $\lim_{r\to\infty} r^2V(r = \infty)$, when the spectral density of the continuous spectrum is conserved.