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

SIGMA 7 (2011), 018, 24 pages      arXiv:1102.5162      https://doi.org/10.3842/SIGMA.2011.018
Contribution to the Proceedings of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design”

Planarizable Supersymmetric Quantum Toboggans

Miloslav Znojil
Nuclear Physics Institute ASCR, 250 68 Rez, Czech Republic

Received November 30, 2010, in final form February 21, 2011; Published online February 25, 2011

Abstract
In supersymmetric quantum mechanics the emergence of a singularity may lead to the breakdown of isospectrality between partner potentials. One of the regularization recipes is based on a topologically nontrivial, multisheeted complex deformations of the line of coordinate x giving the so called quantum toboggan models (QTM). The consistent theoretical background of this recipe is briefly reviewed. Then, certain supersymmetric QTM pairs are shown exceptional and reducible to doublets of non-singular ordinary differential equations a.k.a. Sturm-Schrödinger equations containing a weighted energy E→EW(x) and living in single complex plane.

Key words: supersymmetry; Schrödinger equation; complexified coordinates; changes of variables; single-complex-plane images of Riemann surfaces.

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References

  1. Cooper F., Khare A., Sukhatme U., Supersymmetry and quantum mechanics, Phys. Rep. 251 (1995), 267-385, hep-th/9405029.
  2. Jevicki A., Rodrigues J.P., Singular potentials and supersymmetry breaking, Phys. Lett. B 146 (1984), 55-58.
  3. Junker G., Supersymmetric methods in quantum and statistical physics, Text 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.
  4. Das A., Pernice S.A., Supersymmetry and singular potentials, Nuclear Phys. B 561 (1999), 357-384, hep-th/9905135.
  5. Buslaev V., Grecchi V., Equivalence of unstable anharmonic oscillators and double wells, J. Phys. A: Math. Gen. 26 (1993), 5541-5549.
    Andrianov A.A., Ioffe M.V., Cannata F., Dedonder J.-P., SUSY quantum mechanics with complex superpotentials and real energy spectra, Internat. J. Modern Phys. A 14 (1999), 2675-2688, quant-ph/9806019.
    Cannata F., Junker G., Trost J., Schrödinger operators with complex potential but real spectrum, Phys. Lett. A 246 (1998), 219-226, quant-ph/9805085.
    Bender C.M., Boettcher S., Meisinger P.M., PT-symmetric quantum mechanics, J. Math. Phys. 40 (1999), 2201-2229, quant-ph/9809072.
  6. Bender C.M., Boettcher S., Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80 (1998), 5243-5246, physics/9712001.
  7. Znojil M., Conservation of pseudo-norm in PT symmetric quantum mechanics, math-ph/0104012.
  8. Mostafazadeh A., Pseudo-Hermiticity versus PT Symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian, J. Math. Phys. 43 (2002) 205-214, math-ph/0107001.
    Mostafazadeh A., Pseudo-Hermiticity versus PT-symmetry. II. A complete characterization of non-Hermitian Hamiltonians with a real spectrum, J. Math. Phys. 43 (2002), 2814-2816, math-ph/0110016.
    Mostafazadeh A., Pseudo-Hermiticity versus PT-symmetry. III. Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries, J. Math. Phys. 43 (2002), 3944-3951, math-ph/0203005.
  9. Bender C.M., Brody D.C., Jones H.F., Complex extension of quantum mechanics, Phys. Rev. Lett. 89 (2002), 270402, 4 pages, Erratum, Phys. Rev. Lett. 92 (2004), 119902, 1 page, quant-ph/0208076.
  10. Dorey P., Dunning C., Tateo R., Spectral equivalences, Bethe ansatz equations, and reality properties in PT-symmetric quantum mechanics, J. Phys. A: Math. Gen. 34 (2001), 5679-5704, hep-th/0103051.
  11. Znojil M., Comment on: "Supersymmetry and singular potentials" by Das and Pernice [Nuclear Phys. B 561 (1999), 357-384], Nuclear Phys. B 662 (2003), 554-562, hep-th/0209262.
  12. Dorey P., Dunning C., Tateo R., The ODE/IM correspondence, J. Phys. A: Math. Theor. 40 (2007), R205-R283, hep-th/0703066.
  13. Bender C.M., Making sense of non-Hermitian Hamiltonians, Rep. Progr. Phys. 70 (2007), 947-1018, hep-th/0703096.
  14. Mostafazadeh A., Pseudo-Hermitian representation of quantum mechanics, Int. J. Geom. Methods Mod. Phys. 7 (2010), 1191-1306, arXiv:0810.5643.
  15. Znojil M., Three-Hilbert-space formulation of quantum mechanics, SIGMA 5 (2009), 001, 19 pages, arXiv:0901.0700.
  16. Znojil M., PT-symmetric regularizations in supersymmetric quantum mechanics, J. Phys. A: Math. Gen. 37 (2004), 10209-10222, hep-th/0404145.
  17. Znojil M., PT-symmetric quantum toboggans, Phys. Lett. A 342 (2005), 36-47, quant-ph/0502041.
  18. Fernández F.M., Guardiola R., Ros J., Znojil M., A family of complex potentials with real spectrum, J. Phys. A: Math. Gen. 32 (1999), 3105-3116, quant-ph/9812026.
    Znojil M., Spiked potentials and quantum toboggans, J. Phys. A: Math. Gen. 39 (2006), 13325-13336, quant-ph/0606166.
    Novotný J., http://demonstrations.wolfram.com/TheQuantumTobogganicPaths/.
  19. Znojil M., Jakubský V., Supersymmetric quantum mechanics living on topologically nontrivial Riemann surfaces, Pramana J. Phys. 73 (2009), 397-404, arXiv:0904.2294.
  20. Correa F., Jakubský V., Nieto L.M., Plyushchay M.S., Self-isospectrality, special supersymmetry, and their effect on the band structure, Phys. Rev. Lett. 101 (2008), 030403, 4 pages, arXiv:0801.1671.
    Correa F., Jakubský V., Plyushchay M.S., Finite-gap systems, tri-supersymmetry and self-isospectrality, J. Phys. A: Math. Theor. 41 (2008), 485303, 35 pages, arXiv:0806.1614.
    Siegl P., Supersymmetric quasi-Hermitian Hamiltonians with point interactions on a loop, J. Phys. A: Math. Theor. 41 (2008), 244025, 11 pages.
    Jakubský V., Nieto L.M., Plyushchay M.S., Klein tunneling in carbon nanostructures: a free-particle dynamics in disguise, Phys. Rev. D 63 (2011), 047702, 4 pages, arXiv:1010.0569.
  21. Andrianov A.A., Cannata F., Sokolov A.V., Non-linear supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians. I. General properties, Nuclear Phys. B 773 (2007), 107-136, math-ph/0610024.
  22. Znojil M., Cannata F., Bagchi B., Roychoudhury R., Supersymmetry without hermiticity within PT symmetric quantum mechanics, Phys. Lett. B 483 (2000), 284-289, hep-th/0003277.
  23. Znojil M., PT symmetrized SUSY quantum mechanics, Czechoslovak J. Phys. 51 (2001), 420-428, hep-ph/0101038.
    Znojil M., PT-symmetry and supersymmetry, in GROUP 24: Physical and Mathematical Aspects of Symmetries (Paris, July 15-20, 2002), IOP Publishing, Bristol, 2003, 629-632, hep-th/0209062.
  24. Znojil M., Non-Hermitian SUSY and singular, PT-symmetrized oscillators, J. Phys. A: Math. Gen. 35 (2002), 2341-2352, hep-th/0201056.
  25. Levai G., Znojil M., The interplay of supersymmetry and PT symmetry in quantum mechanics: a case study for the Scarf II potential, J. Phys. A: Math. Gen. 35 (2002), 8793-8804, quant-ph/0206013.
    Sinha A., Roy P., Generation of exactly solvable non-Hermitian potentials with real energies, Czechoslovak J. Phys. 54 (2004), 129-138, quant-ph/0312089.
  26. Caliceti E., Graffi S., Maioli M., Perturbation theory of odd anharmonic oscillators, Comm. Math. Phys. 75 (1980), 51-66.
    Sibuya Y., Global theory of second order linear differential equation with polynomial coefficient, North Holland, Amsterdam, 1975.
    Fernández F.M., Guardiola R., Ros J., Znojil M., Strong-coupling expansions for the PT-symmetric oscillators V(r) = aix+ b(ix)2+ c(ix)3, J. Phys. A: Math. Gen. 31 (1998), 10105-10112.
  27. Znojil M., PT-symmetric harmonic oscillators, Phys. Lett. A 259 (1999), 220-223.
  28. Znojil M., PT-symmetric square well, Phys. Lett. A 285 (2001), 7-10, quant-ph/0101131.
    Quesne C., Bagchi B., Mallik S., Bíla H., Jakubský V., Znojil M., PT-supersymmetric partner of a short-range square well, Czechoslovak J. Phys. 55 (2005), 1161-1166, quant-ph/0507246.
  29. Albeverio S., Fei S.-M., Kurasov P., Gauge fields, point interactions and few-body problems in one dimension, Rep. Math. Phys. 53 (2004), 363-370, quant-ph/0406158.
  30. Znojil M., Tater M., Complex Calogero model with real energies, J. Phys. A: Math. Gen. 34 (2001), 1793-1803, quant-ph/0010087.
    Znojil M., Low-lying spectra in anharmonic three-body oscillators with a strong short-range repulsion, J. Phys. A: Math. Gen. 36 (2003), 9929-9941, quant-ph/0307239.
    Fring A., Smith M., Antilinear deformations of Coxeter groups, an application to Calogero models, J. Phys. A: Math. Theor. 43 (2010), 325201, 28 pages, arXiv:1004.0916.
  31. Znojil M., Quantum knots, Phys. Lett. A 372 (2008), 3591-3596, arXiv:0802.1318.
  32. Znojil M., Quantum toboggans: models exhibiting a multisheeted PT symmetry, J. Phys. Conf. Ser. 128 (2008), 012046, 12 pages, arXiv:0710.1485.
  33. Wessels G.J.C., A numerical and analytical investigation into non-Hermitian Hamiltonians, Master Thesis, University of Stellenbosch, 2008.
  34. Bíla H., Spectra of PT-symmetric Hamiltonians on tobogganic contours, Pramana J. Phys. 73 (2010), 307-314, arXiv:0905.1498.
  35. Dorey P., Millican-Slater A., Tateo R., Beyond the WKB approximation in PT-symmetric quantum mechanics, J. Phys. A: Math. Gen. 38 (2005), 1305-1331, hep-th/0410013.
  36. Znojil M., Quantum toboggans with two branch points, Phys. Lett. A 372 (2008), 584-590, arXiv:0708.0087.
  37. Znojil M., Classification of oscillators in the Hessenberg-matrix representation, J. Phys. A: Math. Gen. 27 (1994), 4945-4968.
  38. Znojil M., Siegl P., Levai G., Asymptotically vanishing PT-symmetric potentials and negative-mass Schrödinger equations, Phys. Lett. A 373 (2009), 1921-1924, arXiv:0903.5468.
  39. 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.
  40. Znojil M., Topology-controlled spectra of imaginary cubic oscillators in the large-L approach, Phys. Lett. A 374 (2010), 807-812, arXiv:0912.1176.
  41. Znojil M., Gemperle F., Mustafa O., Asymptotic solvability of an imaginary cubic oscillator with spikes, J. Phys. A: Math. Gen. 35 (2002), 5781-5793, hep-th/0205181.
  42. Znojil M., Identification of observables in quantum toboggans, J. Phys. A: Math. Theor. 41 (2008), 215304, 14 pages, arXiv:0803.0403.
    Znojil M., Geyer H.B., Sturm-Schrödinger equations: formula for metric, Pramana J. Phys. 73 (2010), 299-306, arXiv:0904.2293.
  43. Znojil M., Re-establishing supersymmetry between harmonic oscillators in D≠1 dimensions, Rend. Circ. Mat. Palermo (2) Suppl. (2003), no. 71, 199-207, hep-th/0203252.
  44. Dieudonné J., Quasi-Hermitian operators, in Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), Pergamon, Oxford, 1961, 115-122.
    Williams J.P., Operators similar to their adjoints, Proc. Amer. Math. Soc. 20 (1969), 121-123.
  45. Langer H., Tretter Ch., A Krein space approach to PT-symmetry, Czechoslovak J. Phys. 54 (2004), 1113-1120.
  46. Lévai G., Znojil M., Systematic search for PT-symmetric potentials with real energy spectra, J. Phys. A: Math. Gen. 33 (2000), 7165-7180.
  47. Günther U., Langer H., Tretter Ch., On the spectrum of the magnetohydrodynamic mean-field α2-dynamo operator, SIAM J. Math. Anal. 42 (2010), 1413-1447, arXiv:1004.0231.
    Znojil M., Günther U., Dynamics of charged fluids and 1/l perturbation expansions, J. Phys. A: Math. Theor. 40 (2007), 7375-7388, math-ph/0610055.
  48. Rüter C.E., Makris K.G., El-Ganainy R., Christodoulides D.N., Segev D.N., Kip D., Observation of parity-time symmetry in optics, Nature Phys. 6 (2010), 192-195.
    Berry M.V., Optical lattices with PT symmetry are not transparent, J. Phys. A: Math. Theor. 41 (2008), 244007, 7 pages.
    Makris K.G., El-Ganainy R., Christodoulides D.N., Musslimani Z.H., Beam dynamics in PT symmetric optical lattices, Phys. Rev. Lett. 100 (2008), 103904, 4 pages.
  49. Hilgevoord J., Time in quantum mechanics, Amer. J. Phys. 70 (2002), 301-306.
  50. Bender C.M., Turbiner A., Analytic continuation of eigenvalue problems, Phys. Lett. A 173 (1993), 442-446.
  51. Mostafazadeh A., Metric operator in pseudo-Hermitian quantum mechanics and the imaginary cubic potential, J. Phys. A: Math. Gen. 39 (2006), 10171-10188, quant-ph/0508195.
  52. Jones H.F., Mateo J., Equivalent Hermitian Hamiltonian for the non-Hermitian −x4 potential, Phys. Rev. D 73 (2006), 085002, 4 pages, quant-ph/0601188.
    Bagchi B., Fring A., Minimal length in quantum mechanics and non-Hermitian Hamiltonian systems, Phys. Lett. A 373 (2009), 4307-4310, arXiv:0907.5354.

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