On the numerical verification of a kernel assumption for the semi-relativistic Hartree equation
We consider the semi-relativistic Hartree equation with an external potential. Recently this equation 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: Connected to the dynamics problem there is an associated energy functional and the assumption is on the size (number of elements) of the kernel of the Hessian of this energy around a solitary wave. In this talk we give indicators to that the assumption is valid. This is related to broken symmetries of the associated energy functional.