The kernel condition of a linearized pseudo-relativistic Hartree equation, a numerical approach

We consider the nonlinear equation i[tp = ('-m)p-(|x|-1*|p|2)p on '3 describing the dynamics of pseudo-relativistic boson stars in the mean-field limit. Recently this equation, with an external potential has been used to describe the dynamics of boson stars under the influence of an external gravitational field. This analysis makes one explicit critical assumption. To the above differential equation we can associate an energy function. The assumption is on the size of the kernel of the Hessian of the energy functional when it is linearized around a soliton. In this paper we provide a numerical indicator that the assumption is satisfied. To achieve this goal, we need to numerically calculate the soliton for a range of normalized frequencies as well as and the spectrum of the linearization around a soliton of the Euler-Lagrange equations describing the minimizer.