Falling of a quantum particle in an inverse square attractive potential

Evolution of a quantum particle in an inverse square potential is studied by analysis of the equation of motion for 〈r2〉. In such a way we identify the conditions of falling of a particle into the center. We demonstrate the existence of a purely quantum limit of falling, namely, a particle does not fall, when the coupling constant is smaller than a certain critical value. Also the time of falling of a particle into the center is estimated. Although there are no stationary energy levels for this potential, we show that there are quasi-stationary states which evolve with 〈r2〉 being constant in time. Our results are compared with measurements of neutral atoms falling in the electric field of a charged wire. Modifications of the experiment, which may help in observing quantum limit of falling, are proposed

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 ≥ 1 in the D = 1 dimensional space, to 1/r m with m ≥ 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