Abstract. This manuscript provides a self-contained introduction to mathematical
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 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 Schr¨odinger 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.
Keywords and phrases. Schr¨odinger operators, quantum mechanics, unbounded
operators, spectral theory.