The authors want to recover the radial potential in the eigenvalue problem
starting from solutions of the form , where are spherical coordinates and are spherical harmonics.
Of course, the family of eigenvalue problems for
where , is highly overdetermined. Consequently, the authors consider the problem of recovering in (1) when a family of spectral data is given. Moreover, they introduce the nonlinear operator
denoting the solution to (1) satisfying the normalization condition
The main purpose of the paper consists of showing that the linearization of problem (2) around uniquely determines a small potential . As a consequence, the authors must show that the kernel of the linearized operator, given by
coincides with , where are the spherical Bessel functions with standard normalization. Moreover, from the asymptotic relationships for the ’s it follows .
Assuming , the assertion will be implied by the (possible) completeness in of the set , where . However, the authors limit themselves to showing the completeness of the previous set when is replaced with , the leading term in its asymptotic expansion.
Finally, the authors show that is complete in if . Numerical examples are also provided.