A numerical study of the Schrödinger - Newton equation 1 : Perturbing the spherically - symmetric stationary states

In this article we consider the linear stability of the spherically-symmetric stationary solutions of the Schrödinger-Newton equations. These have been found numerically by Moroz et al [9] and Bernstein et al [3]. The ground state, characterised as the state of lowest energy , turns out to be linearly stable, with only imaginary eigenvalues. The (n + 1)-th state is linearly unstable having n quadruples of complex eigenvalues (as well as imaginary eigenvalues), where a quadruple consists of {λ, ¯ λ, −λ, − ¯ λ} for complex λ.