  We study the spectrum of a large system of $N$ identical bosons interacting
via a two-body potential with strength $1/N$. In this mean-field regime,
Bogoliubov's theory predicts that the spectrum of the $N$-particle Hamiltonian
can be approximated by that of an effective quadratic Hamiltonian acting on
Fock space, which describes the fluctuations around a condensed state.
Recently, Bogoliubov's theory has been justified rigorously in the case that
the low-energy eigenvectors of the $N$-particle Hamiltonian display complete
condensation in the unique minimizer of the corresponding Hartree functional.
In this paper, we shall justify Bogoliubov's theory for the high-energy part of
the spectrum of the $N$-particle Hamiltonian corresponding to (non-linear)
excited states of the Hartree functional. Moreover, we shall extend the
existing results on the excitation spectrum to the case of non-uniqueness
and/or degeneracy of the Hartree minimizer. In particular, the latter covers
the case of rotating Bose gases, when the rotation speed is large enough to
break the symmetry and to produce multiple quantized vortices in the Hartree
minimizer.
