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.

A momentum representation treatment of the hydrogen atom problem with a generalized uncertainty relation,which leads to a minimal length (∆X i ) min = √ 3β + β ′ , is presented. We show that the distance squared operator can be factorized in the case β ′ = 2β. We analytically solve the s-wave bound-state equation. The leading correction to the energy spectrum caused by the minimal length depends on √ β. An upper bound for the minimal length is found to be about 10 −9fm. * Electronic address: djamilbouaziz@mail.univ-jijel.dz

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.