Domains and Sobolev Spaces associated with Schrödinger Operators with Highly Singular Potentials

This thesis investigates Schrödinger operators with high singular potentials, such as the inverse square and delta potentials. It focuses on the characterization of operator domains and the development of perturbed Sobolev spaces suited for singular interactions. New results include the explicit description of the domain of the squared inverse-square Schrödinger operator and an alternative construction of the delta interaction operator via Helmholtz resolvent limits. Additionally, the thesis extends Strichartz estimates to novel fractional perturbed Sobolev spaces in the L^p case, allowing the study of well-posedness for nonlinear Schrödinger equations with point interactions in dimensions two and three with contraction method.