Abstract
New polynomials related to the mean-field plus isovector pairing model are constructed based on the Stieltjes correspondence. It is shown that there is an one-to-one correspondence between zeros of the two related extended Heine-Stieltjes polynomials satisfying coupled differential equations and the solutions of the Bethe ansatz equations for the model. Similar to the standard pairing among like valence nucleons, an electrostatic interpretation of the location of zeros of the two polynomials is provided. As examples of the solution, the two polynomials related to the solution for three pairs over \(j=1/2,\,3/2,\,5/2\) orbits within the O(5) seniority-zero subspace are provided explicitly, which shows that the number of possible configurations of equilibrium positions of the charges in the two separate systems equals exactly to the number of energy levels in the mean-field plus isovector pairing model.
Similar content being viewed by others
Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors’ comment: The associated data have been included in the article. A Mathematica code used is available upon request.]
References
E. Piasetzky, M. Sargsian, L. Frankfurt, M. Strikman, J.W. Watson, Phys. Rev. Lett. 97, 162504 (2006)
O. Hen et al., (CLAS Collaboration). Science 346, 614 (2014)
S. Frauendorf, A.O. Macchiavelli, Prog. Part. Nucl. Phys. 78, 24 (2014)
C. Andreoiu, C.E. Svensson, A.V. Afanasjev, R.A.E. Austin, M.P. Carpenter, D. Dashdorj, P. Finlay, S.J. Freeman, P.E. Garrett, J. Greene, G.F. Grinyer, A. Görgen, B. Hyland, D. Jenkins, F. Johnston-Theasby, P. Joshi, A.O. Machiavelli, F. Moore, G. Mukherjee, A.A. Phillips, W. Reviol, D.G. Sarantites, M.A. Schumaker, D. Seweryniak, M.B. Smith, J.J. Valiente-Dobón, R. Wadsworth, Phys. Rev. C 75, 041301(R) (2007)
I. Bentley, S. Frauendorf, Phys. Rev. C 88, 014322 (2013)
K.T. Hecht, Phys. Rev. 139, B794 (1965)
K.T. Hecht, Nucl. Phys. 63, 214 (1965)
J.N. Ginocchio, Nucl. Phys. 74, 321 (1965)
F. Pan, Y. He, Y. Wu, Y. Wang, K.D. Launey, J.P. Draayer, Phys. Rev. C 102, 044306 (2020)
F. Pan, C. Qi, L.-R. Dai, G. Sargsyan, K.D. Launey, J.P. Draayer, EPL 132, 32001 (2020)
F. Pan, J.P. Draayer, Phys. Rev. C 66, 044314 (2002)
J. Dukelsky, V.G. Gueorguiev, P. Van Isacker, S. Dimitrova, B. Errea, S.H. Lerma, Phys. Rev. Lett. 96, 072503 (2006)
G. Szegö, Amer. Math. Soc. Colloq. Publ. Vol. 23 (Amer. Math. Soc., Providence, RI, 1975)
F. Marcellán, A. Martínez-Finkelshtein, P. Martínez-González, J. Comput. Appl. Math. 207, 258 (2007)
F. Pan, L. Bao, L. Zhai, X. Cui, J.P. Draayer, J. Phys. A: Math. Theor. 44, 395305 (2011)
F. Pan, X. Guan, L.-R. Dai, Y. Zhang, J.P. Draayer, Eur. Phys. J. Special Topics 229, 2497 (2020)
R.W. Richardson, Phys. Lett. 3, 277 (1963)
R.W. Richardson, Phys. Lett. 5, 82 (1963)
R.W. Richardson, N. Sherman, Nucl. Phys. 52, 221 (1964)
R.W. Richardson, N. Sherman, Nucl. Phys. 52, 253 (1964)
M. Gaudin, J. Physique 37, 1087 (1976)
X. Guan, K.D. Launey, M. Xie, L. Bao, F. Pan, J.P. Draayer, Phys. Rev. C 86, 024313 (2012)
X. Guan, K.D. Launey, M. Xie, L. Bao, F. Pan, J.P. Draayer, Comput. Phys. Commun. 185, 2714 (2014)
I. Scherbak, A theorem of Heine-Stieltjes, the Schubert calculus, and Bethe vectors in the \(sl_p\) Gaudin model, arXiv:math/0211377
E. Mukhin, V. Schechtman, V. Tarasov, A. Varchenko, On the new form of Bethe ansatz equations and separation of variables in the \(sl_3\) Gaudin model, arXiv:math/0609428
F. Pan, X. Ding, K.D. Launey, J.P. Draayer, Nucl. Phys. A 974, 86 (2018)
F. Pan, B. Li, Y.-Z. Zhang, J.P. Draayer, Phys. Rev. C 88, 034305 (2013)
J. Dukelsky, C. Esebbag, S. Pittel, Phys. Rev. Lett. 88, 062501 (2002)
L. Amico, A. Di Lorenzo, A. Mastellone, A. Osterloh, Ann. Phys. 299, 228 (2002)
A. Anfossi, A. LeClair, G. Sierra, J. Stat. Mech. (2005). https://doi.org/10.1088/1742-5468/2005/05/P05011
W.V. Pogosov, J. Phys.: Condens. Matter 24, 075701 (2012)
A. Faribault, O. El Araby, C. Strater, V. Gritsev, Phys. Rev. B 83, 235124 (2011)
I. Marquette, J. Links, J. Stat. Mech. (2012). https://doi.org/10.1088/1742-5468/2012/08/P08019
Acknowledgements
One of the authors (F. P.) is grateful to Prof. Jorge Dukelsky for the lectures on the problem and his suggestion on carrying out this work through many helpful exchanges. Support from the National Natural Science Foundation of China (11675071), the Liaoning Provincial Universities Overseas Training Program (2019GJWYB024), the U.S. National Science Foundation (OIA-1738287 and PHY-1913728), and the LSU-LNNU joint research program with modest but important collaboration-maintaining support from the Southeastern Universities Research Association.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Mark Caprio.
Rights and permissions
About this article
Cite this article
Pan, F., He, Y., Li, A. et al. Extended Heine-Stieltjes polynomials related to the isovector pairing model. Eur. Phys. J. A 57, 218 (2021). https://doi.org/10.1140/epja/s10050-021-00535-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epja/s10050-021-00535-3