  This survey is based on a series of lectures given during the \emph{School on
Random Schr\"odinger Operators} and the \emph{International Conference on
Spectral Theory and Mathematical Physics} at the Pontificia Universidad
Catolica de Chile, held in Santiago in November 2014. As the title suggests,
the presented material has two foci: Harmonic analysis, more precisely, unique
continuation properties of several natural function classes and Schr\"odinger
operators, more precisely properties of their eigenvalues, eigenfunctions and
solutions of associated differential equations. It mixes topics from (rather)
pure to (rather) applied mathematics, as well as classical questions and
results dating back a whole century to very recent and even unpublished ones.
The selection of material covered is based on the selection made for the
minicourse, and is certainly a personal choice corresponding to the research
interests of the authors.
