The Lie Algebraic Structure of Differential Operators Admitting Invariant Spaces of Polynomials

Advances in Applied Mathematics

Volumn 20, Issue 3, 1998, Pages 300-322

  Finkel, Frederico ^a   Kamran, Niky ^b  

^a UNIVERSIDAD COMPLUTENSE DE MADRID   (Spain)

^b Flds Inst for Res in Math Sci   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001328860     PISSN: 01968858     EISSN: None     Source Type: Journal    
DOI: 10.1006/aama.1997.0577     Document Type: Article

References (25)

1. Brihaye Y., Kosinski P. Quasi exactly solvable 2 × 2 matrix equations. J. Math. Phys. 35:1994;3089-3098.
2. Brihaye Y., Giller S., Kosinski P. Linear operators with invariant polynomial space and graded algebra. J. Math. Phys. 38:1997;989-999.
3. Calogero F. Ground state of a one-dimensionalN. J. Math. Phys. 10:1969;2197-2200.
4. 2. J. Algebra. 24:1973;551-564.
5. F. Finkel, A. González-López, M. A. Rodríguez, Quasi-exactly solvable spin 1/2 Schrödinger operators, J. Math. Phys.
6. González-López A., Kamran N., Olver P. J. Quasi-exactly solvable Lie algebras of first order differential operators in two complex variables. J. Phys. A24:1991;3995-4008.
7. González-López A., Kamran N., Olver P. J. Normalizability of one-dimensional quasi-exactly solvable Schrödinger operators. Commun. Math. Phys. 153:1993;117-146.
8. González-López A., Hurtubise J., Kamran N., Olver P. J. Quantification de la cohomologie des algèbres de Lie de champs de vecteurs et fibrés en droites sur des surfaces complexes compactes. C. R. Acad. Sci. (Paris), Série I, 316:1993;1307-1312.
9. González-López A., Kamran N., Olver P. J. New quasi-exactly solvable Hamiltonians in two dimensions. Commun. Math. Phys. 159:1994;503-537.
10. González-López A., Kamran N., Olver P. J. Real Lie algebras of differential operators and quasi-exactly solvable potentials. Philos. Trans. R. Soc. London. A354:1996;1165-1193.
11. Iachello F., Levine R. D. Algebraic Theory of Molecules. 1995;Oxford University Press.
12. Kamran N., Olver P. J. Lie algebras of differential operators and Lie-algebraic potentials. J. Math. Anal. Appl. 145:1990;342-356.
13. Miller W. Jr. Lie Theory and Special Functions. 1968;Academic Press, New York.
14. Minzoni A., Rosenbaum M., Turbiner A. V. Quasi-exactly solvable many-body problems. Mod. Phys. Lett. A11:1996;1977-1984.
15. Post G., Van den Hijligenberg N. gl. Acta Appl. Math. 44:1996;257-268.
16. Post G., Turbiner A. V. Classification of linear differential operators with invariant subspace in monomials. Russ. J. Math. Phys. 3:1995;113-122.
17. Rühl W., Turbiner A. V. Exact solvability of the Calogero and Sutherland models. Mod. Phys. Lett. A10:1995;2213-2221.
18. Scheunert M. The Theory of Lie Superalgebras. 1979;Springer-Verlag.
19. Shifman M. A. New findings in quantum mechanics (partial algebraization of the spectral problem). Int. J. Mod. Phys. A4:1989;2897-2952.
20. Shifman M. A., Turbiner A. V. Quantal problems with partial algebraization of the spectrum. Commun. Math. Phys. 126:1989;347-365.
21. 2. Commun. Math. Phys. 118:1988;467-474.
22. A. V. Turbiner, Lie algebraic approach to the theory of polynomial solutions. I. Ordinary differential equations and finite-difference equations in one variable.
23. A. V. Turbiner, Lie algebraic approach to the theory of polynomial solutions. III. Differential equations in two real variables and general outlook, 1992.
24. Turbiner A. V. Lie algebras, cohomologies, and new findings in quantum mechanics. Contemp. Math. 160:1994;263-310.
25. Ushveridze A. G. Quasi-exactly Solvable Models in Quantum Mechanics. 1994;IOP, Bristol.