Zero energy bound states of singular attractive potentials

Abstract Singular potentials of the form V(r) = –a n /r n , where a n is a positive or negative constant and n > 2 is the integer, have been studied for a long time. Studying these potentials is important because they correspond to a large number of physical systems. The focus of the present paper is on the zero energy states of attractive singular potentials. The existing paradigm is that for negative interaction potentials that vanish as r increases toward infinity, the existence of bound states is possible only for the negative total energy: both classically and quantally, bound states of the zero energy deemed impossible. The first attempt to break this paradigm was made in one of our previous papers for the interaction potential V(r) = –a 3 /r 3 , where a 3 > 0. Two specific examples were neutron-neutron systems and neutron-muon systems for the configuration of the parallel magnetic dipole moments of the two particles in the pair. The existence of zero energy bound states was shown in that paper both for the neutron-neutron system (“neutronium”) and for the neutron-muon system (“neutron-muonic atom”). In the present paper we demonstrate the existence of the bound states of zero energy for all attractive singular potentials V(r) = − a n /r n , where a n > 0, for n > 2. We prove that the corresponding normalization integrals converge and then we actually calculate them explicitly. The final result is the explicit form of the corresponding normalized wave functions of the bound states of zero energy.