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We study the relation between the eigenfrequencies of the Bogoliubovexcitations of Bose-Einstein condensates, and the eigenvalues of the Jacobianstability matrix in a variational approach which maps the Gross-Pitaevskiiequation to a system of equations of motion for the variational parameters. Wedo this for Bose-Einstein condensates with attractive contact interaction in anexternal trap, and for a simple model of a self-trapped Bose-Einsteincondensate with attractive 1/r interaction. The stationary solutions of theGross-Pitaevskii equation and Bogoliubov excitations are calculated using afinite-difference scheme. The Bogoliubov spectra of the ground and excitedstate of the self-trapped monopolar condensate exhibits a Rydberg-likestructure, which can be explained by means of a quantum defect theory. On thevariational side, we treat the problem using an ansatz of time-dependentcoupled Gaussians combined with spherical harmonics. We first apply this ansatzto a condensate in an external trap without long-range interaction, andcalculate the excitation spectrum with the help of the time-dependentvariational principle. Comparing with the full-numerical results, we find agood agreement for the eigenfrequencies of the lowest excitation modes witharbitrary angular momenta. The variational method is then applied to calculatethe excitations of the self-trapped monopolar condensates, and theeigenfrequencies of the excitation modes are compared.

physical review. a, atomic, molecular, and optical physics/physical review, a, atomic, molecular, and optical physics

Relation between the eigenfrequencies of Bogoliubov excitations of  Bose-Einstein condensates and the eigenvalues of the Jacobian in a  time-dependent variational approach