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

A study of new solvable few body problems.
IPNO Participation: M. Lassaut, R.J. Lombard
Collaboration : A. Bachkhaznadji, Laboratoire de Physique Théorique, Département de Physique,
Université Mentouri, Constantine, Algérie
Nous avons étudié des modèles solubles à petit nombre de nucléons. Nous avons généralisé les problèmes
à 3 corps de Calogero et Calogero-Marchiero-Wolfes en introduisant des potentiels à trois corps non
invariants par translation. Après avoir séparé les variables radiales et angulaires par des transformations
de coordonnées nous avons fourni les solutions propres de l'équation de Schrödinger avec le spectre en
énergie correspondant. Nous avons mis en évidence un domaine de la constante de couplage pour lequel
les solutions irrégulières sont de carré intégrables.
The study of exactly solvable non trivial quantum
systems of few interacting particles still retains attention.
The early works of Calogero [1], Sutherland
[2] and Wolfes [3] have been followed by the systematic
classification of Olshanetsky and Perelomov
[4]. Generalizations and new cases have been investigated
in the recent years. In a non exhaustive
way, we quote, for instance, the three-body version
of Sutherland problem, with only a three-body potential,
solved by Quesne [5]. By using supersymmetric
quantum mechanics, Khare and co-workers
gave examples of algebraically solvable three-body
problems of Calogero type in D=1 dimensional
space, with additional translationally invariant twoand/or
three-body potentials[6]. A new integrable
model of the Calogero type, with a non translationally
invariant two-body potential, was worked out in
D=1 by Diaf, Kerris, Lassaut et Lombard [7], and
extended to D-dimensional space by Bachkhazndji,
Lassaut and Lombard [8]. A generalization of the
latter model in D=1 was solved by Meljanac and coworkers
[9], by emphasizing the underlying conformal
SU(1,1) symmetry. However, for the three-body
case and D=1, these authors give only the energy
spectrum and the form of the radial wave function.
The present work investigates again the problem of
Meljanac and co-workers for three particles in the
D=1 dimensional space. The model may be viewed
as a generalization of the three-body Calogero problem
with an additional non-translationally invariant
three-body
potential. We recall here that this model belongs to
the class possessing the underlying conformal SU
(1,1) symmetry. It may also be understood as describing
a system of three light interacting particles
of the same mass m in the harmonic field generated
by a fourth infinitely heavy particle.
We provide the full wavefunction in terms of the radial
and two angular variables, together with the
corresponding eigenvalues. An emphasis is put on
the irregular solutions stressing the domain of the
coupling constants for which the irregular solutions
are physically acceptable. Finally, we also give the
exact results of two other generalizations of the Calogero-Marchioro-Wolfes
three-body problem.
References:
[1] F. Calogero, J. Math. Phys.10(1969) 2191, J.
Math. Phys.12 (1971) 419.
[2] B. Sutherland, J. Math. Phys. 12 (1971) 246,
Phys. Rev. A 4 (1971) 2019.
[3] J. Wolfes,J. Math. Phys. 15 (1974) 1420.
[4] M.A. Olshanetsky and A.M. Perelomov, Phys.
Rep. 71 (1981) 314, Phys. Rep.94 (1983) 6.
[5]C.Quesne, Phys. Rev. A 55 (1997)3931.
[6] A. Khare and R.K. Bhaduri, J. Phys A: Math.
Gen. 27 (1994) 2213.
[7] A. Diaf, A.T. Kerris, M. Lassaut and R.J.
Lombard, J. Phys. A: Math. Gen. 39 (2006) 7305.
[8] A. Bachkhaznadji, M. Lassaut and R.J.
Lombard, J. Phys. A: Math. Theo. 40 (2007) 8791.
[9]S. Meljanac, A. Samsarov, B. Basu-Mallick and
K.S. Gupta Eur. Phys. J. C 49 (2007) 875.
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