Geodesic
From Quantum $A_N$ (Calogero) to $H_4$ (Rational) Model
Symmetry, integrability and geometry: methods and applications, Tome 7 (2011) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A brief and incomplete review of known integrable and (quasi)-exactly-solvable quantum models with rational (meromorphic in Cartesian coordinates) potentials is given. All of them are characterized by $(i)$ a discrete symmetry of the Hamiltonian, $(ii)$ a number of polynomial eigenfunctions, $(iii)$ a factorization property for eigenfunctions, and admit $(iv)$ the separation of the radial coordinate and, hence, the existence of the 2nd order integral, $(v)$ an algebraic form in invariants of a discrete symmetry group (in space of orbits).
Keywords: (quasi)-exact-solvability; rational models; algebraic forms; Coxeter (Weyl) invariants, hidden algebra. 
@article{SIGMA_2011_7_a70,
     author = {Alexander V. Turbiner},
     title = {From {Quantum~}$A_N$ {(Calogero)} to~$H_4$ {(Rational)} {Model}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2011},
     volume = {7},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a70/}
}
TY  - JOUR
AU  - Alexander V. Turbiner
TI  - From Quantum $A_N$ (Calogero) to $H_4$ (Rational) Model
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2011
VL  - 7
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a70/
LA  - en
ID  - SIGMA_2011_7_a70
ER  - 
%0 Journal Article
%A Alexander V. Turbiner
%T From Quantum $A_N$ (Calogero) to $H_4$ (Rational) Model
%J Symmetry, integrability and geometry: methods and applications
%D 2011
%V 7
%U http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a70/
%G en
%F SIGMA_2011_7_a70
Alexander V. Turbiner. From Quantum $A_N$ (Calogero) to $H_4$ (Rational) Model. Symmetry, integrability and geometry: methods and applications, Tome 7 (2011). http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a70/

[1] Evans N. W., “Group theory of the Smorodinsky–Winternitz system”, J. Math. Phys., 32 (1991), 3369–3375 | DOI | MR | Zbl

[2] Turbiner A. V., “Quasi-exactly-solvable problems and the sl(2) group”, Comm. Math. Phys., 118 (1988), 467–474 | DOI | MR | Zbl

[3] Calogero F., “Solution of a three-body problem in one dimension”, J. Math. Phys., 10 (1969), 2191–2196 ; Calogero F., “Solution of the one-dimensional $N$-body problem with quadratic and/or inversely quadratic pair potentials”, J. Math. Phys., 12 (1971), 419–436 | DOI | MR | DOI | MR

[4] Rühl W., Turbiner A., “Exact solvability of the Calogero and Sutherland models”, Modern Phys. Lett. A, 10 (1995), 2213–2221, arXiv: hep-th/9506105 | DOI | MR

[5] Minzoni A., Rosenbaum M., Turbiner A., “Quasi-exactly-solvable many-body problems”, Modern Phys. Lett. A, 11 (1996), 1977–1984, arXiv: hep-th/9606092 | DOI | MR | Zbl

[6] Olshanetsky M. A., Perelomov A. M., “Quantum integrable systems related to Lie algebras”, Phys. Rep., 94 (1983), 313–404 | DOI | MR

[7] Oshima T., “Completely integrable systems associated with classical root systems”, SIGMA, 3 (2007), 061, 50 pp., arXiv: math-ph/0502028 | DOI | MR | Zbl

[8] Brink L., Turbiner A., Wyllard N., “Hidden algebras of the (super) Calogero and Sutherland models”, J. Math. Phys., 39 (1998), 1285–1315, arXiv: hep-th/9705219 | DOI | MR | Zbl

[9] Wolfes J., “On the three-body linear problem with three-body interaction”, J. Math. Phys., 15 (1974), 1420–1424 | DOI | MR

[10] Lie S., “Gruppenregister”, Gesammelte Abhandlungen, v. 5, B. G. Teubner, Leipzig, 1924, 767–773

[11] González-López A., Kamran N., Olver P. J., “Quasi-exactly-solvable Lie algebras of the first order differential operators in two complex variables”, J. Phys. A: Math. Gen., 24 (1991), 3995–4008 ; González-López A., Kamran N., Olver P. J., “Lie algebras of differential operators in two complex variables”, Amer. J. Math., 114 (1992), 1163–1185 | DOI | MR | Zbl | DOI | MR | Zbl

[12] Boreskov K. G.,Turbiner A. V., Lopez Vieyra J. C., “Solvability of the Hamiltonians related to exceptional root spaces: rational case”, Comm. Math. Phys., 260 (2005), 17–44, arXiv: hep-th/0407204 | DOI | MR | Zbl

[13] Tremblay F., Turbiner A. V., Winternitz P., “An infinite family of solvable and integrable quantum systems on a plane”, J. Phys. A: Math. Theor., 42 (2009), 242001, 10 pp., arXiv: 0904.0738 | DOI | MR | Zbl

[14] García M. A. G., Turbiner A. V., “The quantum $H_3$ integrable system”, Internat. J. Modern Phys. A, 25 (2010), 5567–5594, arXiv: 1007.0737 | DOI | MR | Zbl

[15] García M. A. G., Turbiner A. V., “The quantum $H_4$ integrable system”, Modern Phys. Lett. A, 26 (2011), 433–447, arXiv: 1011.2127 | DOI | MR | Zbl

[16] García M. A. G., Los Sistemas Integrables $H_3$ y $H_4$, PhD Thesis, UNAM, 2011 (in Spanish)