A numerical study of the Schrödinger-Newton equations
A numerical study of the Schrödinger-Newton equations
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DOI：
10.1088/0951-7715/16/1/307
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发表时间：
2003
期刊：
影响因子：
1.7
通讯作者：
R. Harrison;I. Moroz;K. P. Tod
中科院分区：
文献类型：
--
作者：
R. Harrison;I. Moroz;K. P. Tod
The Schrodinger-Newton (S-N) equations were proposed by Penrose [18] as a model for gravitational collapse of the wave-function. The potential in the Schrodinger equation is the gravity due to the density of $|\psi|^2$, where $\psi$ is the wave-function. As with normal Quantum Mechanics the probability, momentum and angular momentum are conserved. We first consider the spherically symmetric case, here the stationary solutions have been found numerically by Moroz et al [15] and Jones et al [3]. The ground state which has the lowest energy has no zeros. The higher states are such that the $(n+1)$th state has $n$ zeros. We consider the linear stability problem for the stationary states, which we numerically solve using spectral methods. The ground state is linearly stable since it has only imaginary eigenvalues. The higher states are linearly unstable having imaginary eigenvalues except for $n$ quadruples of complex eigenvalues for the $(n+1)$th state, where a quadruple consists of $\{\lambda,\bar{\lambda},-\lambda,-\bar{\lambda}\}$. Next we consider the nonlinear evolution, using a method involving an iteration to calculate the potential at the next time step and Crank-Nicolson to evolve the Schrodinger equation. To absorb scatter we use a sponge factor which reduces the reflection back from the outer boundary condition and we show that the numerical evolution converges for different mesh sizes and time steps. Evolution of the ground state shows it is stable and added perturbations oscillate at frequencies determined by the linear perturbation theory. The higher states are shown to be unstable, emitting scatter and leaving a rescaled ground state. The rate at which they decay is controlled by the complex eigenvalues of the linear perturbation. Next we consider adding another dimension in two different ways: by considering the axisymmetric case and the 2-D equations. The stationary solutions are found. We modify the evolution method and find that the higher states are unstable. In 2-D case we consider rigidly rotationing solutions and show they exist and are unstable.
