Continuous non-relativistic bound states

Consider a group of charged point(at least considered as such in this non-relativistic limit) particles such electrons,protons
, nucleii alone in an empty infinite universe and NOT considering any internal structure nor spin nor magnetic interaction though can consider spin as far as symmetry, antisymmetry
requirements such belonging to a certain Young diagram. And also assume atleast one pair of particles has opposite charge eg not all particles have the same sign of charge.
Consider only coulomb potential and normal non-relativistic momentum. Has anyone seen a proof that for a group of charges of which it may be considered no bound states
exist, which may be somewhat controversial, that the bound states are only continuous and there is no finite increment no matter how small but NOt zero between the whole system
eigenstates. By bound states here I mean all states which have a NEGATIVE energy eigenvalue of which there definitely MUST be obviously. Here i am considering the zero point
of energy as that of particles an infinite(in the limit) distance apart eg of infinitesimal wave function density at rest.
For example such a zero point energy could also(and in fact more clearly obvious) be the case of a group of particles all of the same charge sign. I think you know what I mean.