Matrix Quasi-Exactly Solvable Jacobi Elliptic Hamiltonian


We construct a new example of 2 × 2-matrix quasi-exactly solvable (QES) Hamiltonian which is associated to a potential depending on the Jacobi elliptic functions. We establish three necessary and sufficient algebraic conditions for the previous operator to have an invariant vector space whose generic elements are polynomials. This operator is called quasi-exactly solvable.

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A. Nininahazwe, "Matrix Quasi-Exactly Solvable Jacobi Elliptic Hamiltonian," Open Journal of Microphysics, Vol. 3 No. 3, 2013, pp. 53-59. doi: 10.4236/ojm.2013.33010.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] A. V. Turbiner, “Quasi-Exactly-Solvable Problems and sl(2) Algebra,” Communications in Mathematical Physics, Vol. 118, No. 3, 1988, pp. 467-474. doi:10.1007/BF01466727
[2] A. G. Ushveridze, “Quasi-Exactly Solvable Models in Quantum Mechanics,” Institute of Physics Publishing, 1995.
[3] A. V. Turbiner, “Lame Equation sl(2) Algebra and Isospectral Deformations,” Journal of Physics A: Mathematical and General, Vol. 22, 1989, pp. 1-144. doi:10.1088/0303-4470/22/1/001
[4] M. A. Shifman and A. V. Turbiner, “Quantal Problems with Partial Algebraization of the Spectrum,” Communications in Mathematical Physics, Vol. 126, No. 2, 1989, pp. 347-365. doi:10.1007/BF02125129
[5] A. González-López, N. Kamran and P. J. Olver, “Normalizability of One-Dimensional Quasi-Exactly Solvable Schr?dinger Operators,” Communications in Mathematical Physics, Vol. 153, No. 1, 1993, pp. 117-146. doi:10.1007/BF02099042
[6] A. González-López, N. Kamran and P. J. Olver, “Quasi-Exactly Solvable Lie Algebras of Differential Operators in Two Complex Variables,” Journal of Physics A, Vol. 24, No. 17, 1991, p. 3995. doi:10.1088/0305-4470/24/17/016
[7] R. Zhdanov, “Quasi-Exactly Solvable Matrix Models,” Physics Letters B, Vol. 405, No. 3-4, 1997, pp. 253-256. doi:10.1016/S0370-2693(97)00655-2
[8] Y. Brihaye and P. Kosinski, “Quasi Exactly Solvable Matrix Models in sl(n),” Physics Letters B, Vol. 424, No. 1-2, 1997, pp. 43-47. doi:10.1016/S0370-2693(98)00167-1
[9] Y. Brihaye, A. Nininahazwe and B. P. Mandal, “PT-Symmetric, Quasi-Exactly Solvable Matrix Hamiltonians,” Journal of Physics A: Mathematical and Theoretical, Vol. 40, No. 43, 2007, pp. 13063-13073 doi:10.1088/1751-8113/40/43/014
[10] Y. Brihaye and A. Nininahazwe, “Extended Jaynes-Cummings models and (Quasi)-Exact Solvability,” Journal of Physics A: Mathematical and Theoretical, Vol. 39, No. 33, 2006, pp. 1-14. doi:10.1088/0305-4470/39/31/011
[11] Y. Brihaye and B. Hartmann, “Quasi-Exactly Solvable N × N-Matrix Schrodinger Operators,” Modern Physics Letters A, Vol. 16, No. 29, 2001, pp. 1895-1906. doi:10.1142/S0217732301005242
[12] Y. Brihaye and M. Godard, “Quasi Exactly Solvable Extensions of the Lamé Equation,” Journal of Mathematical Physics, Vol. 34, No. 11, 1993, p. 5283. doi:10.1063/1.530304

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