A numerical study of the Schr dinger Newton equations

In this paper we present a numerical study of the Schrödinger–Newton (SN) equations. We begin by considering the linear stability of the spherically symmetric stationary solutions found numerically by Moroz et al (1998 Class. Quantum Grav. 15 2733–42) and Bernstein et al (1998 Mod. Phys. Lett. A13 2327–36). The ground state, characterized 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,−} for complex λ. Then, we consider the time-dependent SN equations in three-dimensions with three kinds of symmetry: spherically symmetric, axially symmetric and translationally symmetric. We find that the solutions show a balance between the dispersive tendencies of the Schrödinger equation and the gravitional attraction from the Poisson equation. Only the ground state is stable, and lumps of probability attract each other gravitationally before dispersing.