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 Schrodinger 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.