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.

We study the radial Schrödinger equation for a particle of mass m in the field of the inverse-square potential α/r 2 in the medium-weak-coupling region, i.e., with −1/4 2mα/ 2 3/4. By using the renormalization method of Beane et al., with two regularization potentials, a spherical square well and a spherical δ shell, we illustrate that the procedure of renormalization is independent of the choice of the regularization counterterm. We show that, in the aforementioned range of the coupling constant α, there exists at most one bound state, in complete agreement with the method of self-adjoint extensions. We explicitly show that this bound state is due to the attractive square-well and delta-function counterterms present in the renormalization scheme. Our result for 2mα/ 2 = −1/4 is in contradiction with some results in the literature.

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The Kratzer's potential V (r) = g 1 /r 2 − g 2 /r is studied in quantum mechanics with a generalized uncertainty principle, which includes a minimal length (∆X) min = √ 5β. In momentum representation, the Schrödinger equation is a generalized Heun's differential equation, which reduces to a hypergeometric and to a Heun's equations in special cases. We explicitly show that the presence of this finite length regularizes the potential in the range of the coupling constant g 1 where the corresponding Hamiltonian is not self-adjoint. In coordinate space, we perturbatively derive an analytical expression for the bound states spectrum in the first order of the deformation parameter β. We qualitatively discuss the effect of the minimal length on the vibration-rotation energy levels of diatomic molecules, through the Kratzer interaction. By comparison with an experimental result of the hydrogen molecule, an upper bound for the minimal length is found to be of about 0.01Å.We argue that the minimal length would have some physical importance in studying the spectra of such systems. * Electronic address: djamilbouaziz@univ-jijel.dz

The problem of a particle of mass m in the field of the inverse square potential α/r 2 is studied in quantum mechanics with a generalized uncertainty principle, characterized by the existence of a minimal length. Using the coordinate representation, for a specific form of the generalized uncertainty relation, we solve the deformed Schrödinger equation analytically in terms of confluent Heun functions. We explicitly show the regularizing effect of the minimal length on the singularity of the potential. We discuss the problem of bound states in detail and we derive an expression for the energy spectrum in a natural way from the square integrability condition; the results are in complete agreement with the literature.

The pseudoharmonic oscillator potential is studied in non relativistic quantum mechanics with a generalized uncertainty principle characterized by the existence of a minimal length scale, (∆x) min = √ 5β. By using a perturbative approach, we derive an analytical expression of the energy spectrum in the first order of the minimal length parameter β. We investigate the effect of this fundamental length on the vibration-rotation energy levels of diatomic molecules through this potential function interaction. We explicitly show that the minimal length would have some physical importance in studying the spectra of diatomic molecules. * Electronic address: djamilbouaziz@univ-jijel.dz † Electronic address: abdelmalek-boukhellout@univ-jijel.dz