Bound States of Energy Dependent Singular Potentials

We consider attractive power-law potentials depending on energy through their coupling constant. These potentials are proportional to 1/|x| (m) with m a parts per thousand yen 1 in the D = 1 dimensional space, to 1/r (m) with m a parts per thousand yen 2 in the D = 3 dimensional space. We study the ground state of such potentials. First, we show that all singular attractive potentials with an energy dependent coupling constant are bounded from below, contrarily to the usual case. In D = 1, a bound state of finite energy is found with a kind of universality for the eigenvalue and the eigenfunction, which become independent on m for m > 1. We prove the solution to be unique. A similar situation arises for D = 3 for m > 2, except that, in this case, the solution is not directly comparable to a bound state: the wave function, though square integrable, diverges at the origin.