Parametrizations of the Poisson-Schr\"odinger Equations in Spherical Symmetry

TL;DR

This paper explores different parametrizations of static solutions to the Poisson-Schrödinger equations in spherical symmetry, providing explicit formulas and numerical analysis to relate various parameter choices and their physical interpretations.

Contribution

It introduces new explicit formulas relating different parametrizations of static states in the Poisson-Schrödinger system, enhancing understanding of their physical and geometrical properties.

Findings

Minimum value of the discrete node variable for the ground state identified

Explicit formulas relating various parametrizations derived

Numerical inversion techniques used to analyze parameter transformations

Abstract

We consider the asymptotically flat standing wave solutions to the Poisson-Schr\"{o}dinger system of equations known as static states. These solutions can be parameterized 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.