Mathematics

Through the Kreĭn-Višik-Birman extension scheme, unlike the previous
classical analysis based on von Neumann's theory, we reproduce the construction
and classification of all self-adjoint realisations of two intimately related models:
the three-dimensional hydrogenoid-like Hamiltonians with singular perturbation
supported at the centre (the nucleus), and the Schördinger operators on the halfline
with Coulomb potentials centred at the origin. These two problems are technically
equivalent, albeit sometimes treated by their own in the the literature. Based
on such scheme, we then recover the formula to determine the eigenvalues of each
self-adjoint extension, which are corrections to the non-relativistic hydrogenoid energy
levels.We discuss in which respect the Kreĭn-Višik-Birman scheme is somehow
more natural in yielding the typical boundary condition of self-adjointness at the
centre of the perturbation and in identifying the eigenvalues of each extension.