Inverse problem of reconstructing a confining potential for the radial Schrödinger equation

The problem of reconstructing a confining (increasing at infinity) potential for the radial Schrödinger equation from the spectral distribution function is considered. A perturbation to the potential that changes the first $n$ levels and normalization constants is constructed and its asymptotic behavior as $r\to\infty$ investigated. The connection between the moments of the spectral distribution function and the derivatives of the potential at the origin is established. The procedure for reconstructing an increasing potential from a finite set of experimental data is proposed.