Python code for Schrodinger-Newton solitons with axial symmetry

Abstract

These Python codes solve the Schrodinger-Newton problem of a nonrelativistic particle bound in a gravitational well created by the probability distribution of its own location. Rather than being restricted to the traditional assumption of spherical symmetry, these codes allow axial symmetry. Two methods are considered. One uses cylindrical coordinates to represent the Schrodinger and gravitational Poisson equations as two-dimensional partial differential equations which are then solved by finite-difference methods of matrix diagonalization and SOR iterations, respectively. The other method uses spherical coordinates and a partial wave analysis, which leads to coupled ordinary differential equations also solved by finite-difference methods. The second method is faster and much more stable with respect to convergence for simultaneous solutions of the Schrodinger and Poisson equations. The first method is used as a check on the second.