We study the problem of the attractive inverse square potential in quantum mechanics with a generalized uncertainty relation. Using the momentum representation, we show that this potential is regular in this framework. We solve analytically the s-wave bound states equation in terms of Heun's functions. We discuss in detail the bound states spectrum for a specific form of the generalized uncertainty relation. The minimal length may be interpreted as characterizing the dimension of the system.

We study the radial Schroedinger equation for a particle of mass m in the field of a singular attractive α/r 2 potential with 2mα > 1/4. This potential is relevant to the fabrication of nanoscale atom optical devices, is said to be the potential describing the dipole-bound anions of polar molecules, and is the effective potential underlying the universal behavior of three-body systems in nuclear physics and atomic physics, including aspects of Bose-Einstein condensates, first described by Efimov. New results in three-body physical systems motivate the present investigation. Using the regularization method of Beane et al., we show that the corresponding "renormalization group flow" equation can be solved analytically. We find that it exhibits a limit cycle behavior and has infinitely many branches. We show that a physical meaning for self-adjoint extensions of the Hamiltonian arises naturally in this framework.

We solve analytically the Schrödinger equation for the N-dimensional inverse square potential in quantum mechanics with a minimal length in terms of Heun's functions. We apply our results to the problem of a dipole in a cosmic string background. We find that a bound state exists only if the angle between the dipole moment and the string is larger than π/4. We compare our results with recent conflicting conclusions in the literature. The minimal length may be interpreted as a radius of the cosmic string.

We solve the problem of a neutral atom interacting with an charged wire, giving rise to an attractive 1/r 2 potential in two dimensions. We show how a suitable average over all possible self-adjoint extensions of the radial Schroedinger Hamiltonian eventually leads to the classical formula for absorption of the atom, a formula shown to be in agreement with a recent experiment.

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