Preface

Overview

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

upon.

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.

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