Sobolev Spaces and Unitary Groups

This lecture covers the proof that initial data in H^2 solve the free Schrödinger equation in the L^2-sense, the definition of the generator of a one-parameter unitary group, and the properties of the Laplacian as the generator of free time evolution. It also introduces the momentum operator as the generator of the translation group on the real line, defines Sobolev spaces on an interval, presents the baby Sobolev embedding theorem, and discusses the translation group on L^2 over an interval.