arXiv subjects

Michel Bawin

Publications and source records attributed to Michel Bawin.

4 recordsLinked to original sources

arXiv2014

Singular inverse-square potential: renormalization and self-adjoint extensions for medium to weak coupling

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 $-1/4\leq2mα/\hbar^{2}\leq3/4$. By using the renormalization method of Beane \textit{et} \textit{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 $2mα/\hbar^{2}=-1/4$ is in contradiction with some results in the literature.

math-ph↗

arXiv2010

Singular inverse square potential in arbitrary dimensions with a minimal length: Application to the motion of a dipole in a cosmic string background

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.

quant-ph↗

arXiv2007

Regularization of the Singular Inverse Square Potential in Quantum Mechanics with a Minimal length

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.

quant-ph↗

arXiv2004

Renormalization of the singular attractive $1/r^4$ potential

We study the radial Schrödinger equation for a particle of mass $m$ in the field of a singular attractive $g^2/{r^4}$ potential with particular emphasis on the bound states problem. Using the regularization method of Beane \textit{et al.}, we solve analytically the corresponding ``renormalization group flow" equation. We find in agreement with previous studies that its solution exhibits a limit cycle behavior and has infinitely many branches. We show that a continuous choice for the solution corresponds to a given fixed number of bound states and to low energy phase shifts that vary continuously with energy. We study in detail the connection between this regularization method and a conventional method modifying the short range part of the potential with an infinitely repulsive hard core. We show that both methods yield bound states results in close agreement even though the regularization method of Beane \textit{et al.} does not include explicitly any new scale in the problem. We further illustrate the use of the regularization method in the computation of electron bound states in the field of neutral polarizable molecules without dipole moment. We find the binding energy of s-wave polarization bound electrons in the field of C$_{60}$ molecules to be 17 meV for a scattering length corresponding to a hard core radius of the size of the molecule radius ($\sim 3.37$ Å). This result can be further compared with recent two-parameter fits using the Lennard-Jones potential yielding binding energies ranging from 3 to 25 meV.

quant-ph↗