A numerical study of the Schrodinger-Newton equation 1: Perturbing the   spherically-symmetric stationary states
  We consider the linear stability of the spherically-symmetric stationary solutions of the Schrodinger-Newton equations. We find that the ground state is linearly stable, with only imaginary eigenvalues, while the n-th excited state has n quadruples of complex eigenvalues as well as purely imaginary ones and so is linearly unstable. 
R. Harrison, I. Moroz and K.P. Tod