Parametrizations of the Poisson–Schrödinger equations in spherical symmetry

We consider the asymptotically flat standing wave solutions to the Poisson–Schrödinger system of equations. These equations are also known as the Schrödinger–Newton equations and are the Newtonian limit of the Einstein–Klein–Gordon equations. The asymptotically flat standing wave solutions to the Poisson–Schrödinger equations are known as static states. These solutions can be parametrized using a variety of choices of two continuous parameters and one discrete parameter, each having a useful physical-geometrical interpretation. The values of the discrete variable determines the number of nodes (zeros) in the solution. We use numerical inversion techniques to analyze transformations between various informative choices of parametrization by relating each of them to a standard set of three parameters. Based on our computations, we propose explicit formulas for these relationships. Our computations also show that for the standard choice of continuous variables, the zero-node ground state yields a minimum value of a geometrically natural discrete variable. We give an explicit formula for this minimum value. We use these results to confirm two related observations from previous work by the author and others, and suggest additional applications and approaches to understand these phenomena analytically.