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

The inverse problem in the case of bound states.
IPNO Participation: R.J. Lombard
Collaboration : R. Yekken and F.-Z. Ighezou, Institut de Physique, USTHB, Bab Ezzouar, Alger, Algeria
Nous étudions le problème inverse dans le cas des états liés dans l’espace à 3 dimensions. Le potentiel est
supposé local et à symétrie sphérique. Notre méthode est basée sur des relations reliant les moments de la
densité de l’état fondamental à l’énergie la plus basse de chaque excitation de moment angulaire L. La
reconstruction de la densité à partir de ses moments est effectuée par l’intermédiaire de sa transformée de
Fourier. La partie à grand transfert est obtenue par des approximants de Padé. Le potentiel est déterminé
avec une précision qui dépend du nombre de moments connus. La solution est unique dans la mesure où
une infinité de moments sont connus. En pratique la connaissance des 15-20 premiers moments est suffisante
pour une précision de l’ordre de 1 %. La méthode est appliquée dans trois cas de spectres.
Lz
Studying the inverse problem in quantum mechanics
goes back to the forties. Closely related to the
description of spectra and scattering in terms of
the Schrödinger equation, it addresses the question
of how and to what extend the data determine
the potential. The literature on this subject is numerous.
Among the pioneering works, the most
frequently quoted are those of Bargmann,
Marchenko, Jost and Kohn, Gel'fand and Levitan.
For a more detailed review, we refer the reader to
the textbooks of Newton, Chadan and Sabatier
and Grosse and Martin.
Most of the works were dedicated to scattering
data, with or without accounting for bound states.
As far as spectra are considered, the usual procedure
was to postulate a parametrized expression
to the potential, and find the best parameters by a
fit to the energies. The question of a more systematic
approach was raised with the adventure of
heavy quark systems. Thacker, Quigg and Rosner
developed a method based on solitons together
with the Korteweg-de Vries equation. Application of
their method to few simple examples gives an idea
of the accuracy which can be reached as function
of the soliton number.
An iterative method has been proposed by Sacks.
Also based on the Gel'fand-Levitan solution, it
uses spectral data to generate the impulse response
function for given boundary conditions.
Our approach is different. It is based on the generalization
of an inequality derived by Bertlmann and
Martin long ago. The latter connects the mean
square radius of the ground state density to the
lowest dipole excitation energy. The generalization
has been derived and tested in previous articles. It
gives access to the moments of the ground state
density in terms of the excitation energy of the lowest
state of each angular momentum. In the present
work, we show that it provides us with a sys-
tematic way for solving the inverse problem in the
case of bound states. In particular it specifies the
conditions for the uniqueness of the solution.
The method is applied to three spectra :
E(L) = - 1/( L + 1)
E(L) = (L + 1) 1/2
E(L) = L(L + 1).
References:
V. Bargmann, Rev. Mod. Phys. 21, 488 (1949) ;
Phys Rev 75, 301 (1949).
V.A. Marchenko, Dokl. Akad. Nauk SSSR 72, 457
(1950) ; 104, 695 (1955).
R. Jost and W. Kohn, Phys. Rev. 87, 979 (1952) ;
88, 382 (1952); K. Dan. Vidensk. Selsk. Mat-Fys,
Medd. 27, N* 9 (1953).
I.M. Gel'fand and B.M. Levitan, Am. Math. Soc.
Trans. 1, 253 (1955).
R. Newton, Scattering Theory of Waves and Particles,
2 nd edition, Springer-Verlag, New York, Heidelberg,
Berlin, (1982).
K. Chadan and P.C. Sabatier, Inverse Problems in
Quantum Scattering Theory, 2nd edition, Springer-
Verlag (1989).
H. Grosse and A. Martin, Particle Physics and the
Schrödinger equation, Cambridge University Press
(1997).
H.B. Thacker, C. Quigg and J.L. Rosner, Phys.
Rev D 18, 274 (1978).
P.E. Sacks, Inverse Problem 4 (1988) 1055.
R.A. Bertlmann and A. Martin, Nucl. Phys. B 168,
111
(1980).
F.-Z. Ighezou and R.J. Lombard, Ann. Phys. (NY)
278, 265 (1999).
R. Mezhoud, F.-Z. Ighezou, A. Chouchaoui, A.T.
Kerris and R.J. Lombard, Ann. Phys. (NY), 308,
143 (2003).
93