  We study the relation between the eigenfrequencies of the Bogoliubov
excitations of Bose-Einstein condensates, and the eigenvalues of the Jacobian
stability matrix in a variational approach which maps the Gross-Pitaevskii
equation to a system of equations of motion for the variational parameters. We
do this for Bose-Einstein condensates with attractive contact interaction in an
external trap, and for a simple model of a self-trapped Bose-Einstein
condensate with attractive 1/r interaction. The stationary solutions of the
Gross-Pitaevskii equation and Bogoliubov excitations are calculated using a
finite-difference scheme. The Bogoliubov spectra of the ground and excited
state of the self-trapped monopolar condensate exhibits a Rydberg-like
structure, which can be explained by means of a quantum defect theory. On the
variational side, we treat the problem using an ansatz of time-dependent
coupled Gaussians combined with spherical harmonics. We first apply this ansatz
to a condensate in an external trap without long-range interaction, and
calculate the excitation spectrum with the help of the time-dependent
variational principle. Comparing with the full-numerical results, we find a
good agreement for the eigenfrequencies of the lowest excitation modes with
arbitrary angular momenta. The variational method is then applied to calculate
the excitations of the self-trapped monopolar condensates, and the
eigenfrequencies of the excitation modes are compared.
