Transactions in Theoretical and Mathematical Physics

Transactions in Theoretical and Mathematical Physics

Lie Algebraic Approaches to Advanced Few-Body Hamiltonians in Physics

Document Type : Original Article

Author
Department of Physics, Qom University of Technology, Qom, Iran
Abstract
In quantum many-body systems, the emergence of complex interactions generally prevents the straightforward application of conventional analytical methods such as direct solution of the Schrödinger equation, variation theory, or the WKB approximation. An alternative algebraic perspective—employing techniques such as representation theory, reduction transformations, and gauge transformations—provide successful ways by identifying the hidden Lie algebra underlying a given problem.
On the other hand, few-body systems remain challenging, as their symmetries are limited rather than N-body cases. In this work, we demonstrate that by incorporating geometric insight, one may establishes a meaningful connection between the mathematical structure and the physical content of the problem.
Keywords
Subjects

1. J. J. Sakurai, Modern Quantum Mechanics, (Addison
Wesley, 1985)#

2. F. Calogero, Solution of the one-dimensional N-body
problems Problems with Quadratic and/or Inversely
Quadratic Pair Potentials, J. Math. Phys. 12, 419 (1971).
DOI: https://doi.org/10.1063/1.1665604#

3. B. Sutherland, Exact Results for a Quantum Many-Body
Problem in OneDimension, Phys. Rev. A5,1372(1972).
DOI: https://doi.org/10.1103/PhysRevA.5.137
2#

4. R. Gilmore, Lie Groups, Physics and Geometry, (Cam
bridge University Press, 2008)#

5. F. Pan and J. P. Draayer, Exact Solutions for Some Nu
clear Many-Body Problems, Ann. Phys. 271, 120 (1999).
DOI: https://doi.org/10.1006/aphy.1998.5870#

6. J. F. Cariñena, J. de Lucas, A. Ramos, A geometric ap
proach to time evolution operators of Lie quantum sys
tems, Int. J. Theor. Phys. 48, 1379 (2009). DOI: https:
//doi.org/10.1007/s10773-009-9911-0#

7. F. Calogero, Classical Many Body Problems Amenable
to Exact Treatments in One, Two and Three Dimensional
Space, (Springer-Verlag Berlin Heidelberg, 2001)#

8. M. A. Olshanetsky and A. M. Perelomov, Quantum Inte
grable Systems Related to Lie Algebras, Phys. Rep. 94,
313 (1983). DOI: https://doi.org/10.1016/0370-1573(83)90018-2#

9. A. V. Turbiner, Quasi Exactly Solvable Problems and
SL(2) Algebra, Commun. Math. Phys. 118, 467 (1988).
DOI: https://doi.org/10.1007/BF01466727#

10. A.G.López, N. KamranandP.J. Olver, New Quasi Ex
actly Solvable Hamiltonians in Two Dimensions, Com
mun. Math. Phys. 159, 503 (1994). DOI: https://doi.
org/10.1007/BF02100490#

11. D. Gómez-Ullate, N. Kamran, R. Milson, An Extension
of Bochner’s problem: Exceptional Invariant Subspaces,
J. Approx. Theory 162, 987 (2010). DOI: https://do
i.org/10.1016/j.jat.2009.11.002#

12. F. Brauneis et al., Comparison of renormalized inter
actions using one-dimensional few-body systems as a
testbed, Phys. Rev. A 111, 013303 (2025). DOI: https:
//doi.org/10.1103/PhysRevA.111.013303#

13. I. G. Macdonald, Symmetric Functions and Hall Poly
nomials, 2nd edition, (Clarendon Press, Oxford, 1995)#

14. H. Rahmati, A. Latifi, Exact solution of the two and
three-body interactions in the trigonometric three-body
problem via Jack polynomials, Few-Body Syst. 60, 1
(2019). DOI: https://doi.org/10.1007/s00601-019-1464-5#

15. H. Rahmati, Lie algebraic approach to the Hellmann
Hamiltonian by considering perturbation method, Theor.
Math. Phys. 221, 2144 (2024). DOI: https://doi.or
g/10.1134/S0040577924120121"

16. A. N. Kolmogorov, On conservation of conditionally
periodic motions for a small change in Hamilton’s func
tion, Dokl. Akad. Nauk SSSR 98, 527 (1954)#

17. M. Hamzavi, K. E. Thylwe, and A. A. Rajabi, Ap
proximate bound states solution of the Hellmann poten
tial, Commun. Theor. Phys. 60, 1 (2013). DOI: https:
//doi.org/10.1088/0253-6102/60/1/01#

18. S. M. Ikhdair and R. Sever, A perturbative treatment for
the bound states of the Hellmann potential, J. Mol. Struct.
809, 103 (2007). DOI: https://doi.org/10.1016/
j.molstruc.2006.06.028#
 
Volume 3, Issue 1
Winter 2026
Pages 20-24

  • Receive Date 22 January 2026
  • Accept Date 22 February 2026
  • First Publish Date 22 February 2026
  • Publish Date 01 February 2026