Spectral and scattering properties of point interactions

Abstract

Point interactions in all relevant dimensions are important to model short rangeinteractions in quantum mechanics. They are exactly solvable and play an important role in solid state, nuclear and atomic physics. In this thesis, we investigate point interactions on flat space and on a general compact manifold in two and three dimensions.The problem contains divergences and they can be renormalized using the couplingconstant renormalization method. We derive the properties of the bound states, provethat the resolvent operator that we have found defines a Hamiltonian, and calculatescattering cross sections. We repeat some of our calculations on a general compactRiemannian manifold and solve the same problem on a sphere and hyperbolic planesH2 and H3. At the end we give some numerical examples.