**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.

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