exotic nuclei structure and reaction noyaux exotiques ... - IPN - IN2P3

A pseudo Coulombian potential in D=1 dimensional space.
IPNO Participation: R.J. Lombard
Collaboration : R. Mezhoud, Faculty of Sciences, Boumerdes University, Boumerdes, Algerie, and
R. Yekken, Institut de Physique, USTHB, Bab Ezzouar, Alger, Algeria.
Dans l’espace à 1 dimension, nous avons étudié les états liés du potentiel V(x) =-e/x + b/x 2 où e et b sont
des constantes positives. Ces états apparaissent sur le demi-plan droit. Dans la limite b tendant vers 0,
nous retrouvons le spectre du potentiel Coulombien. Les propriétés de supersymétrie sont discutées. Le
modèle est étendu en considérant des constantes de couplage complexes. Nous étudions également des
effets non-linéaires introduits par une dépendance en énergie de la constante e.
The study of Coulomb-like potentials in the D=1
dimensional space goes back to the early works of
Flügge and Marshall, Loudon and Haines and
Roberts. A short historical background have been
given by Reyes and del Castillo-Mussot. These
authors remarked that the -1/|x| and -1/(|x| + a)
potentials have been widely considered. On the
contrary, the -1/x has not been much investigated,
although it is of interest to describe asymmetric
wells in connection with semiconductors and insulators.
Reyes and del Castillo-Mussot used
Laplace and Fourier transforms to obtain the solutions
corresponding to the -1/x and -1/|x| potentials.
The -1/x potential case was recently reexamined
by Yangqiang Ran et al, who gave the correct
answer for the spectrum. Let us quote also that the
regularization of the one-dimensional Coulomb
potential has been made by Nouicer by using the
principle of minimal lengths.
In a different context, Calogero and Marchioro
found that a potential of the form
Σ i