Mathematical methods in quantum mechanics - Schrodinger operators.pdf

Mathematical Methods
in Quantum Mechanics
With Applications to Schr¨odinger Operators
Gerald Teschl
Gerald Teschl
Fakult¨atf¨urMathematik
Nordbergstraße 15
Universit¨atWien
1090 Wien, Austria
E-mail address: Gerald.******@
URL: ./~gerald/
2000 Mathematics subject classification. 81-01, 81Qxx
Abstract. This manuscript provides a self-contained introduction to math-
ematical methods in quantum mechanics (spectral theory) with applications
to Schr¨odinger operators. The first part covers mathematical foundations
of quantum mechanics from self-adjointness, the spectral theorem, quantum
dynamics (including Stone’s and the RAGE theorem) to perturbation theory
for self-adjoint operators.
The second part starts with a detailed study of the free Schr¨odinger op-
erator respectively position, momentum and angular momentum operators.
Then we develop Weyl-Titchmarsh theory for Sturm-Liouville operators and
apply it to spherically symmetric problems, in particular to the hydrogen
atom. Next we investigate self-adjointness of atomic Schr¨odinger operators
and their essential spectrum, in particular the HVZ theorem. Finally we
have a look at scattering theory and prove pleteness in the
short range case.
Keywords and phrases. Schr¨odinger operators, quantum mechanics, un-
bounded operators, spectral theory.
Typeset by AMS-LATEX and Makeindex.
Version: June 23, 2005
Copyright
c 1999-2005 by Gerald Teschl
Contents
Preface vii
Part 0. Preliminaries
Chapter 0. A first look at Banach and Hilbert spaces 3
§. Warm up: Metric and topological spaces 3
§. The Banach space of continuous functions 10
§. The geometry of Hilbert spaces 14
§. Completeness 19
§. Bounded operators 20
§. Lebesgue Lp spaces 22
§. Appendix: The uniform boundedness principle 27
Part 1. Mathematical Foundations of Quantum Mechanics
Chapter 1. Hilbert spaces 31
§. Hilbert spaces 31
§. Orthonormal bases 33
§. The projec