On the two-dimensional Coulomb-like potential with a central point interaction

In the first part of the paper, we introduce the Hamiltonian -\Delta -Z/\sqrt{x^{2}+y^{2}}, Z > 0, as a self-adjoint operator in L^{2}(\ {R}^{2}). A general central point interaction combined with the two-dimensional Coulomb-like potential is constructed and the properties of the resulting one-parameter family of Hamiltonians are studied in detail. The construction is also reformulated in the momentum representation and a relation between the coordinate and the momentum representation is derived. In the second part of the paper, we prove that the two-dimensional Coulomb-like Hamiltonian can be derived as a norm resolvent limit of the Hamiltonian of a Hydrogen atom in a planar slab as the width of the slab tends to zero.