Geometric spectral inversion for singular potentials

The function E = F(v) expresses the dependence of a discrete eigenvalue E of the Schroedinger Hamiltonian H = -\Delta + vf(r) on the coupling parameter v. We use envelope theory to generate a functional sequence \{f^{[k]}(r)\} to reconstruct f(r) from F(v) starting from a seed potential f^{[0]}(r). In the power-law or log cases the inversion can be effected analytically and is complete in just two steps. In other cases convergence is observed numerically. To provide concrete illustrations of the inversion method it is first applied to the Hulth\'en potential, and it is then used to invert spectral data generated by singular potentials with shapes of the form f(r) = -a/r + b\sgn(q)r^q and f(r) = -a/r + b\ln(r), a, b > 0. For the class of attractive central potentials with shapes f(r) = g(r)/r, with g(0)< 0 and g'(r)\ge 0, we prove that the ground-state energy curve F(v) determines f(r) uniquely.