On a family of finitely many point interactions Hamiltonians free of ultraviolet pathologies in two and three dimensions

In this thesis, we investigate Hamiltonians with point interaction potentials in two and three dimensions under the assumption of exchange symmetry with respect to the point positions. We examine the entire family of many-center point interaction Hamiltonians using von Neumann's theory of self-adjoint extensions. We demonstrate that a significant subfamily of point interaction Hamiltonians remains well-defined and non-trivial when scattering centers coincide. We apply this framework to the three-body problem with two heavy particles and one light particle within a formal Born-Oppenheimer approximation. In three dimensions, this approximation amounts to having a regular Hamiltonian at the origin, and the Efimov effect manifests itself when the two-body interactions reach the unitary limit, where the \(S\)-wave scattering length is infinite. We trace the Efimov spectrum away from the unitary limit, providing a consistent qualitative picture and partly also a quantitative picture for cases with finite scattering lengths. In two dimensions, we introduce a novel function to analyze the effective potential in the Born-Oppenheimer approximation. Moreover, we provide an estimate for the ground state energy of the three-body system.