  Bogoliubov Theory provides important predictions for the low energy
properties of the weakly interacting Bose gas. Recently, Bogoliubov's
predictions could be justified rigorously in \cite{BBCS3} for translation
invariant systems in the Gross-Pitaveskii regime, where $N$ bosons in $\Lambda
= [0;1]^3\subset \mathbb{R}^3$ interact through a potential whose scattering
length is of size $ N^{-1}$. In this note, we review recent results from
\cite{BSS2}, a joint work with B. Schlein and S. Schraven, which extends the
analysis of \cite{BBCS3} to systems of bosons in $\mathbb{R}^3$ that are
trapped by an external potential.
