Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 005, 24 pages      arXiv:2010.15276

Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model

Ian Marquette a and Christiane Quesne b
a) School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia
b) Physique Nucléaire Théorique et Physique Mathématique,Université Libre de Bruxelles, Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels, Belgium

Received September 01, 2021, in final form January 03, 2022; Published online January 14, 2022

A shape invariant nonseparable and nondiagonalizable three-dimensional model with quadratic complex interaction was introduced by Bardavelidze, Cannata, Ioffe, and Nishnianidze. However, the complete hidden symmetry algebra and the description of the associated states that form Jordan blocks remained to be studied. We present a set of six operators $\{A^{\pm},B^{\pm},C^{\pm}\}$ that can be combined to build a ${\mathfrak{gl}}(3)$ hidden algebra. The latter can be embedded in an ${\mathfrak{sp}}(6)$ algebra, as well as in an ${\mathfrak{osp}}(1/6)$ superalgebra. The states associated with the eigenstates and making Jordan blocks are induced in different ways by combinations of operators acting on the ground state. We present the action of these operators and study the construction of an extended biorthogonal basis. These rely on establishing various nontrivial polynomial and commutator identities. We also make a connection between the hidden symmetry and the underlying superintegrability property of the model. Interestingly, the integrals generate a cubic algebra. This work demonstrates how various concepts that have been applied widely to Hermitian Hamiltonians, such as hidden symmetries, superintegrability, and ladder operators, extend to the pseudo-Hermitian case with many differences.

Key words: quantum mechanics; complex potentials; pseudo-Hermiticity; Lie algebras; Lie superalgebras.

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  1. Bagchi B.K., Supersymmetry in quantum and classical mechanics, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Vol. 116, Chapman & Hall/CRC, Boca Raton, FL, 2001.
  2. Bardavelidze M.S., Cannata F., Ioffe M.V., Nishnianidze D.N., Three-dimensional shape invariant non-separable model with equidistant spectrum, J. Math. Phys. 54 (2013), 012107, 11 pages, arXiv:1212.4805.
  3. Bender C.M., Introduction to ${\mathcal{PT}}$-symmetric quantum theory, Contemp. Phys. 46 (2005), 277-292, arXiv:quant-ph/0501052.
  4. Bender C.M., Making sense of non-Hermitian Hamiltonians, Rep. Progr. Phys. 70 (2007), 947-1018, arXiv:hep-th/0703096.
  5. Bender C.M., Boettcher S., Real spectra in non-Hermitian Hamiltonians having ${\mathcal{PT}}$ symmetry, Phys. Rev. Lett. 80 (1998), 5243-5246, arXiv:physics/9712001.
  6. Bender C.M., Dunne G.V., Meisinger P.N., dSimdsek M., Quantum complex Hénon-Heiles potentials, Phys. Lett. A 281 (2001), 311-316, arXiv:quant-ph/0101095.
  7. Cannata F., Ioffe M.V., Nishnianidze D.N., New methods for the two-dimensional Schrödinger equation: SUSY-separation of variables and shape invariance, J. Phys. A: Math. Gen. 35 (2002), 1389-1404, arXiv:hep-th/0201080.
  8. Cannata F., Ioffe M.V., Nishnianidze D.N., Exactly solvable nonseparable and nondiagonalizable two-dimensional model with quadratic complex interaction, J. Math. Phys. 51 (2010), 022108, 14 pages, arXiv:0910.0590.
  9. Cannata F., Ioffe M.V., Nishnianidze D.N., Equidistance of the complex two-dimensional anharmonic oscillator spectrum: the exact solution, J. Phys. A: Math. Theor. 45 (2012), 295303, 11 pages, arXiv:1206.4013.
  10. Cooper F., Khare A., Sukhatme U., Supersymmetry and quantum mechanics, Phys. Rep. 251 (1995), 267-385, arXiv:hep-th/9405029.
  11. Frappat L., Sciarrino A., Sorba P., Dictionary on Lie superalgebras, arXiv:hep-th/9607161.
  12. Gendenshtein L.E., Derivation of exact spectra of the Schrödinger equation by means of supersymmetry, JETP Lett. 38 (1983), 356-359.
  13. Ioffe M.V., Cannata F., Nishnianidze D.N., Exactly solvable two-dimensional complex model with a real spectrum, Theoret. and Math. Phys. 148 (2006), 960-967, arXiv:hep-th/0512110.
  14. Junker G., Supersymmetric methods in quantum and statistical physics, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1996.
  15. Marquette I., Quesne C., Ladder operators and hidden algebras for shape invariant nonseparable and nondiagonalizable models with quadratic complex interaction. I. Two-dimensional model, SIGMA 18 (2022), 004, 11 pages, arXiv:2010.15273.
  16. Moshinsky M., Smirnov Yu.F., The harmonic oscillator in modern physics, Harwood, Amsterdam, 1996.
  17. Mostafazadeh A., Pseudo-Hermiticity for a class of nondiagonalizable Hamiltonians, J. Math. Phys. 43 (2002), 6343-6352, arXiv:math-ph/0207009.
  18. Mostafazadeh A., Pseudo-Hermiticity versus $\mathcal{PT}$ symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian, J. Math. Phys. 43 (2002), 205-214, arXiv:math-ph/0107001.
  19. Mostafazadeh A., Erratum: Pseudo-Hermiticity for a class of nondiagonalizable Hamiltonians [J. Math. Phys. 43, 6343 (2002)], J. Math. Phys. 44 (2003), 943, arXiv:math-ph/0301030.
  20. Mostafazadeh A., Pseudo-Hermitian representation of quantum mechanics, Int. J. Geom. Methods Mod. Phys. 7 (2010), 1191-1306, arXiv:0810.5643.
  21. Nanayakkara A., Real eigenspectra in non-Hermitian multidimensional Hamiltonians, Phys. Lett. A 304 (2002), 67-72.
  22. Pauli W., On Dirac's new method of field quantization, Rev. Modern Phys. 15 (1943), 175-207.
  23. Scholtz F.G., Geyer H.B., Hahne F.J.W., Quasi-Hermitian operators in quantum mechanics and the variational principle, Ann. Physics 213 (1992), 74-101.

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