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

The many-body problem with an energy-dependent
confining potential.
IPNO Participation: R.J. Lombard
Collaboration : J. Mareš, Nuclear Physics Institute, 25068 Rež, Czech Republic
Nous considérons un système de N bosons liés par des forces harmoniques à deux corps, dont la fréquence
dépend de l’énergie totale du système. Cette dépendance en énergie produit des propriétés remarquables,
en particulier sur le spectre en énergie. Lorsque les nombres quantiques augmentent, l’énergie totale
du système ne peut dépasser une limite supérieure, qui est indépendantes du nombre de bosons. Par ailleurs,
l’énergie de l’état fondamental augmente avec N. Il en résulte une densité de niveau qui tend rapidement
vers l’infini lorsque N et/ou les nombres quantiques croissent.
We have studied wave equations with potentials
depending on the eigenvalues. Solutions of such
equations exhibit properties quite unusual with respect
to the known solutions of the ordinary
Schrödinger equation for the same potential
shape. This is particularly spectacular in the case
of the harmonic oscillator with a linear energy dependence.
The shape of the spectrum reminds the
Coulomb spectrum, with the ground state significantly
lowered with respect to the other states. The
excitation energy cannot exceed a saturation energy
and the density of states thus becomes very
large for high quantum numbers. This upper bound
to the spectrum is a characteristic of confining potentials.
It depends on the chosen energy dependence.
The energy dependence of the potential arises first
in relativistic quantum mechanics. The Pauli-
Schrödinger equation is a good example. It occurs
also in the relativistic many-body problem treated
through the manifestly covariant formalism.
Energy dependent potentials were successfully
used in the calculation of spectra of the heavy
quark systems : charmonia and bottomia. The present
work is a first step towards the application of
these potentials to 3,4 and 5 quark systems.
Clearly, it is tempting to extend the study of energy
dependent interactions to a many-body system of
N- particles. We will consider two-body harmonic
oscillator interactions which lead to analytically
solvable models.
sider N bosons of equal mass interacting by a twobody
harmonic potential. The oscillator frequency
is chosen to be linearly dependent on the total energy
of the system. Actually, the energy dependence
is not dictated by general principles. It should
be derived from the considered underlying theory.
Our choice is merely motivated by simplicity. It is
sufficient to underline the main features and draw
attention to new aspects of the many-body problem
brought by the energy dependence of the twobody
force.
The general formula for the spectrum is derived. It
is characterized by a saturation energy : as the
quantum numbers increase, the spectrum cannot
exceed a finite upper bound. This saturation energy
is independent of N, while the ground state
energy is increasing with N. Consequently, as N
increases, the density of states tends to infinity,
and the concept of discrete state become dubious.
In this limit, the transition between states cost
practically no energy. Thus, as N increases, the
system is expected to become highly unstable.
References:
J. Formánek, R.J. Lombard and J. Mareš, Czech J.
Phys. 54 (2004) 1143.
R.J. Lombard, J. Mareš and C. Volpe, J. Phys. G:
Nucl. Part. Phys. 34 (2007) 1879.
H. Sazdjan, Ann. Phys. (NY) 191 (1989) 52.
This aspect of the harmonic forces has been used
since a long time in the many-body problem. It has
been applied widely in connection with the
nuclear shell model. It has allowed us to test the
validity of the mean field approximation. More recently,
it has been used to describe the
Bose-Einstein condensate or the relativistic massless
bosons system ; In the present work, we con-
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