Bogoliubov excitation spectrum of trapped Bose gases in the Gross-Pitaevskii regime

We consider an inhomogeneous system of $N$ bosons in ${\mathbb{R}}^3$ confined by an external potential and interacting via a repulsive potential of the form $N^{2}V(N(x-y))$. We prove that the low-energy excitation spectrum of the system is determined by the eigenvalues of an effective one-particle operator, which agrees with Bogoliubov’s approximation.