Stability of gravitationally bounded Bose-Einstein condensate consisting of self-interacting bosons

Considering Dark Matter as light self-interacting bosons, we study the stability of compact gravitationally bounded Bose-Einstein condensates (Bose stars). The problem was considered before under the Cowling approximation, i.e. ignoring variations in the gravitational potential. This approach reduces the eigenvalue problem from a fourth-order equation to a second-order one. The Cowling approximation is not used in this work, and the problem is reduced to a second-order equation for the displacement vector. Eigenfunctions and eigenvalues are found for the radial and the few lowest non-radial perturbations. The perturbations are oscillating, which implies the stability of the condensate.