Exact renormalization group on point interactions

The goal of this thesis is to present a non-perturbative renormalization of point interactions in non-relativistic quantum mechanics using Exact Renormalization Group (ERG) techniques. The analyzed model is a particle interacting with a Dirac-Delta potential on various two dimensional Riemannian Manifolds, namely the Euclid Plane R^2 , the Hyperbolic Plane H^2 , and the Sphere S^2 . In all cases the problem contains logarithmic divergences, hence regularization and renormalization are needed. The renormalization is performed using the ERG method, where one gets an expression for the flow equation of the effective coupling constant by integrating out degrees of freedom between the bare scale \Lambda and the effective scale \lambda. The main conclusion ofthis work is to show that the flow equations are identical in all three manifolds under