Book

Abstract

This manuscript provides a brief introduction to mathematical methods in quantum mechanics (spectral theory) with applications to Schrödinger 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 Schrödinger operator 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 Schrödinger operators and their essential spectrum, in particular the HVZ theorem. Finally we have a look at scattering theory and prove asymptotic completeness in the short range case.

MSC2000: 81-01, 81Qxx, 46-01, 34Bxx, 47B25

Keywords: Schrödinger operators, quantum mechanics, unbounded operators, spectral theory.