  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.
