Exact energy eigenvalues for an attractive Coulomb potential with zero-radius hard core

Path integrals for two‐ and three‐dimensional δ‐function perturbations

“…In two and three dimensions, in setting α = 0 some sort of defect is produced, which one may call a zero-radius hard core. Let us emphasize that again the entire Green function has to be taken into account and not only part of it (compare [50]). This shows once more that it is not enough just to throw away the continuous part of the spectrum (say, in the Coulomb case), but a proper regularization always takes into account the entire Green function yielding inadmissible contributions from both the discrete and continuous spectrum of the unperturbed problem.…”

“…This kind of regularization procedure is known from functional analysis [3] to define the corresponding proper self-adjoint Hamiltonian, and our purpose is to show that it can also be applied to regularize the corresponding path integral formulation. We will find that the entire Green function has to be taken into account in the regularization procedure, and not only part of it (compare [50]). …”