The present text was written for my course Schr¨ odinger Operators held
at the University of Vienna in winter 1999, summer 2002, summer 2005,
and winter 2007. It gives a brief but rather self-contained introduction
to the mathematical methods of quantum mechanics with a view towards
applications to Schr¨ odinger operators. The applications presented are highly
selective; as a result, many important and interesting items are not touched
Part 1 is a stripped-down introduction to spectral theory of unbounded
operators where I try to introduce only those topics which are needed for
the applications later on. This has the advantage that you will (hopefully)
not get drowned in results which are never used again before you get to
the applications. In particular, I am not trying to present an encyclopedic
reference. Nevertheless I still feel that the first part should provide a solid
background covering many important results which are usually taken for
granted in more advanced books and research papers.
My approach is built around the spectral theorem as the central object.
Hence I try to get to it as quickly as possible. Moreover, I do not take the
detour over bounded operators but I go straight for the unbounded case. In
addition, existence of spectral measures is established via the Herglotz rather
than the Riesz representation theorem since this approach paves the way for
an investigation of spectral types via boundary values of the resolvent as the
spectral parameter approaches the real line.