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


SIGMA 6 (2010), 075, 10 pages      arXiv:1009.4764      http://dx.doi.org/10.3842/SIGMA.2010.075
Contribution to the Proceedings of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design”

Supersymmetrical Separation of Variables in Two-Dimensional Quantum Mechanics

Mikhail V. Ioffe
Saint-Petersburg State University, St.-Petersburg, 198504 Russia

Received August 24, 2010, in final form September 19, 2010; Published online September 24, 2010

Abstract
Two different approaches are formulated to analyze two-dimensional quantum models which are not amenable to standard separation of variables. Both methods are essentially based on supersymmetrical second order intertwining relations and shape invariance - two main ingredients of the supersymmetrical quantum mechanics. The first method explores the opportunity to separate variables in the supercharge, and it allows to find a part of spectrum of the Schrödinger Hamiltonian. The second method works when the standard separation of variables procedure can be applied for one of the partner Hamiltonians. Then the spectrum and wave functions of the second partner can be found. Both methods are illustrated by the example of two-dimensional generalization of Morse potential for different values of parameters.

Key words: supersymmetry; separation of variables; integrability; solvability.

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References

  1. Schrödinger E., A method of determining quantum-mechanical eigenvalues and eigenfunctions, Proc. Roy. Irish Acad. Sect. A 46 (1940), 9-16.
    Schrödinger E., Further studies on solving eigenvalue problems by factorization, Proc. Roy. Irish Acad. Sect. A 46 (1940), 183-206.
    Schrödinger E., The factorization of the hypergeometric equation, Proc. Roy. Irish Acad. Sect. A 47 (1941), 53-54.
    Infeld L., Hull T.E., The factorization method, Rev. Modern Phys. 23 (1951), 21-68.
    Dabrowska J.W., Khare A., Sukhatme U., Explicit wavefunctions for shape-invariant potentials by operator techniques, J. Phys. A: Math. Gen. 21 (1988), L195-L200.
  2. Cooper F., Khare A., Sukhatme U., Supersymmetry and quantum mechanics, Phys. Rep. 251 (1995), 267-385, hep-th/9405029.
    Junker G., Supersymmetrical methods in quantum and statistical physics, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1996.
    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.
  3. Witten E., Dynamical breaking of supersymmetry, Nuclear Phys. B 185 (1981), 513-554.
  4. Gendenshtein L.E., Derivation of exact spectra of the Schrödinger equation by means of SUSY, JETP Lett. 38 (1983), 356-359.
  5. Darboux G., Sur un proposition relative aux equations lineares, C. R. Acad. Sci. Paris 94 (1882), 1456-1459.
  6. Andrianov A.A., Borisov N.V., Ioffe M.V., Quantum systems with identical energy spectra, JETP Lett. 39 (1984), 93-97.
    Andrianov A.A., Borisov N.V., Ioffe M.V., Factorization method and Darboux transformation for multidimensional Hamiltonians, Theoret. and Math. Phys. 61 (1984), 1078-1088.
    Andrianov A.A., Borisov N.V., Ioffe M.V., The factorization method and quantum systems with equivalent energy spectra, Supersymmetic equivalent quantum systems, Phys. Lett. A 105 (1984), 19-22.
  7. Bagrov V.G., Samsonov B.F., Darboux transformation, factorization and supersymmetry in one-dimensional quantum mechanics, Theoret. and Math. Phys. 104 (1995), 1051-1060.
  8. Miller W. Jr., Symmetry and separation of variables, Encyclopedia of Mathematics and its Applications, Vol. 4, Addison-Wesley Publishing Co., Reading, Mass. - London - Amsterdam, 1977.
  9. Eisenhart L.P., Enumeration of potentials for which one-particle Schrödinger equations are separable, Phys. Rev. 74 (1948), 87-89.
  10. Calogero F., Solution of one-dimensional N-body problems with quadratic and/or inversely quadratic pair potential, J. Math. Phys. 12 (1971), 419-436.
  11. Turbiner A.V., Quasi-exactly-solvable problems and sl(2) algebra, Comm. Math. Phys. 118 (1988), 467-474.
    Ushveridze A.G., Quasi-exactly solvable models in quantum mechanics, Soviet J. Particles and Nuclei 20 (1989), 504-528.
  12. Cannata F., Ioffe M.V., Roychoudhury R., Roy P., A new class of PT-symmetric Hamiltonians with real spectra, Phys. Lett. A 281 (2001), 305-310, quant-ph/0011089.
    Debergh N., Van Den Bossche B., Samsonov B.F., Darboux transformations for quasi-exactly solvable Hamiltonians, Internat. J. Modern Phys. A 17 (2002), 1577-1587, quant-ph/0201105.
  13. Andrianov A.A., Borisov N.V., Ioffe M.V., Eides M.I., Supersymmetric mechanics: a new look at the equivalence of quantum systems, Theoret. and Math. Phys. 61 (1984), 965-972.
    Andrianov A.A., Borisov N.V., Eides M.I., Ioffe M.V., Supersymmetric origin of equivalent quantum systems, Phys. Lett. A 109 (1985), 143-148.
  14. Andrianov A.A., Ioffe M.V., Pauli fermions as components of D=2 supersymmetrical quantum mechanics, Phys. Lett. B 205 (1988), 507-510.
  15. Ioffe M.V., Neelov A.I., Pauli equation and the method of supersymmetric factorization, J. Phys. A: Math. Gen. 36 (2003), 2493-2506, hep-th/0302004.
  16. Ioffe M.V., Kuru S., Negro J., Nieto L.M., SUSY approach to Pauli Hamiltonians with an axial symmetry, J. Phys. A: Math. Gen. 39 (2006), 6987-7001, hep-th/0603005.
  17. Demircioglu B., Kuru S., Önder M., Verçin A., Two families of superintegrable and isospectral potentials in two dimensions, J. Math. Phys. 43 (2002), 2133-2150, quant-ph/0201099.
  18. Andrianov A.A., Ioffe M.V., Spiridonov V.P., Higher-derivative supersymmetry and the Witten index, Phys. Lett. A 174 (1993), 273-279, hep-th/9303005.
    Andrianov A.A., Ioffe M.V., Cannata F., Dedonder J.-P., Second order derivative supersymmetry, q deformations and the scattering problem, Internat. J. Modern Phys. A 10 (1995), 2683-2702, hep-th/9404061.
  19. Andrianov A.A., Ioffe M.V., Nishnianidze D.N., Polynomial SUSY in quantum mechanics and second derivative Darboux transformations, Phys. Lett. A 201 (1995), 103-110, hep-th/9404120.
    Andrianov A.A., Ioffe M.V., Nishnianidze D.N., Polynomial supersymmetry and dynamical symmetries in quantum mechanics, Theoret. and Math. Phys. 104 (1995), 1129-1140.
    Cannata F., Ioffe M.V., Nishnianidze D.N., Double shape invariance of the two-dimensional singular Morse model, Phys. Lett. A 340 (2005), 31-36, hep-th/0504077.
  20. Ioffe M.V., Valinevich P.A., New two-dimensional quantum models partially solvable by the supersymmetrical approach, J. Phys. A: Math. Gen. 38 (2005), 2497-2510, hep-th/0409153.
    Ioffe M.V., Mateos Guilarte J., Valinevich P.A., Two-dimensional supersymmetry: from SUSY quantum mechanics to integrable classical models, Ann. Physics 321 (2006), 2552-2565, hep-th/0603006.
  21. Ioffe M.V., Negro J., Nieto L.M., Nishnianidze D.N., New two-dimensional integrable quantum models from SUSY intertwining, J. Phys. A: Math. Gen. 39 (2006), 9297-9308.
  22. 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, hep-th/0201080.
  23. Ioffe M.V., A SUSY approach for investigation of two-dimensional quantum mechanical systems, J. Phys. A: Math. Gen. 37 (2004), 10363-10374, hep-th/0405241.
  24. Landau L., Lifshitz E., Quantum mechanics, Pergamon, London, 1965.
  25. Andrianov A.A., Cannata F., Ioffe M.V., Nishnianidze D.N., Systems with higher-order shape invariance: spectral and algebraic properties, Phys. Lett. A 266 (2000), 341-349, quant-ph/9902057.
    Cannata F., Ioffe M.V., Nishnianidze D.N., Double shape invariance of the two-dimensional singular Morse model, Phys. Lett. A 340 (2005), 31-36, hep-th/0504077.
  26. Ioffe M.V., Mateos Guilarte J., Valinevich P.A., A class of partially solvable two-dimensional quantum models with periodic potentials, Nuclear Phys. B 790 (2008), 414-431, arXiv:0706.1344.
  27. Ioffe M.V., Nishnianidze D.N., Exact solvability of a two-dimensional real singular Morse potential, Phys. Rev. A 76 (2007), 052114, 5 pages, arXiv:0709.2960.

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