Advertisement

Journal of High Energy Physics

, 2019:194 | Cite as

Nonlinear symmetries of perfectly invisible PT-regularized conformal and superconformal mechanics systems

  • Juan Mateos Guilarte
  • Mikhail S. PlyushchayEmail author
Open Access
Regular Article - Theoretical Physics
  • 15 Downloads

Abstract

We investigate how the Lax-Novikov integral in the perfectly invisible PT-regularized zero-gap quantum conformal and superconformal mechanics systems affects on their (super)-conformal symmetries. We show that the expansion of the conformal symmetry with this integral results in a nonlinearly extended generalized Shrödinger algebra. The PT-regularized superconformal mechanics systems in the phase of the unbroken exotic nonlinear \( \mathcal{N} \) = 4 super-Poincaré symmetry are described by nonlinearly super-extended Schrödinger algebra with the osp(2|2) sub-superalgebra. In the partially broken phase, the scaling dimension of all odd integrals is indefinite, and the osp(2|2) is not contained as a sub-superalgebra.

Keywords

Conformal and W Symmetry Extended Supersymmetry Integrable Hierarchies 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    V. de Alfaro, S. Fubini and G. Furlan, Conformal Invariance in Quantum Mechanics, Nuovo Cim. A 34 (1976) 569 [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    F. Calogero, Solution of the one-dimensional N body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971) 419 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    V.P. Akulov and A.I. Pashnev, Quantum superconformal model in (1, 2) space, Theor. Math. Phys. 56 (1983) 862 [INSPIRE].CrossRefGoogle Scholar
  4. [4]
    S. Fubini and E. Rabinovici, Superconformal quantum mechanics, Nucl. Phys. B 245 (1984) 17 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    E.A. Ivanov, S.O. Krivonos and V.M. Leviant, Geometry of Conformal Mechanics, J. Phys. A 22 (1989) 345 [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  6. [6]
    E.A. Ivanov, S.O. Krivonos and V.M. Leviant, Geometric superfield approach to superconformal mechanics, J. Phys. A 22 (1989) 4201 [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  7. [7]
    D.Z. Freedman and P.F. Mende, An Exactly Solvable N Particle System in Supersymmetric Quantum Mechanics, Nucl. Phys. B 344 (1990) 317 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    N. Wyllard, (Super)conformal many body quantum mechanics with extended supersymmetry, J. Math. Phys. 41 (2000) 2826 [hep-th/9910160] [INSPIRE].
  9. [9]
    S. Bellucci, A. Galajinsky, E. Ivanov and S. Krivonos, AdS 2 /CFT 1 , canonical transformations and superconformal mechanics, Phys. Lett. B 555 (2003) 99 [hep-th/0212204] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  10. [10]
    S. Bellucci, A. Galajinsky and S. Krivonos, New many-body superconformal models as reductions of simple composite systems, Phys. Rev. D 68 (2003) 064010 [hep-th/0304087] [INSPIRE].ADSGoogle Scholar
  11. [11]
    S. Bellucci, A.V. Galajinsky and E. Latini, New insight into WDVV equation, Phys. Rev. D 71 (2005) 044023 [hep-th/0411232] [INSPIRE].ADSGoogle Scholar
  12. [12]
    A. Galajinsky, O. Lechtenfeld and K. Polovnikov, Calogero models and nonlocal conformal transformations, Phys. Lett. B 643 (2006) 221 [hep-th/0607215] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    A. Galajinsky, O. Lechtenfeld and K. Polovnikov, N = 4 superconformal Calogero models, JHEP 11 (2007) 008 [arXiv:0708.1075] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    S. Krivonos and O. Lechtenfeld, Many-particle mechanics with D(2, 1 : α) superconformal symmetry, JHEP 02 (2011) 042 [arXiv:1012.4639] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  17. [17]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    P. Claus, M. Derix, R. Kallosh, J. Kumar, P.K. Townsend and A. Van Proeyen, Black holes and superconformal mechanics, Phys. Rev. Lett. 81 (1998) 4553 [hep-th/9804177] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    J.A. de Azcarraga, J.M. Izquierdo, J.C. Perez Bueno and P.K. Townsend, Superconformal mechanics and nonlinear realizations, Phys. Rev. D 59 (1999) 084015 [hep-th/9810230] [INSPIRE].ADSMathSciNetGoogle Scholar
  20. [20]
    G.W. Gibbons and P.K. Townsend, Black holes and Calogero models, Phys. Lett. B 454 (1999) 187 [hep-th/9812034] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    J. Michelson and A. Strominger, Superconformal multiblack hole quantum mechanics, JHEP 09 (1999) 005 [hep-th/9908044] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  22. [22]
    G.F. de Teramond, H.G. Dosch and S.J. Brodsky, Baryon Spectrum from Superconformal Quantum Mechanics and its Light-Front Holographic Embedding, Phys. Rev. D 91 (2015) 045040 [arXiv:1411.5243] [INSPIRE].ADSGoogle Scholar
  23. [23]
    S.J. Brodsky, G.F. de Téramond, H.G. Dosch and C. Lorcé, Universal Effective Hadron Dynamics from Superconformal Algebra, Phys. Lett. B 759 (2016) 171 [arXiv:1604.06746] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  24. [24]
    E. D’Hoker and L. Vinet, Dynamical Supersymmetry of the Magnetic Monopole and the 1/r 2 Potential, Commun. Math. Phys. 97 (1985) 391 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  25. [25]
    J. Michelson and A. Strominger, The Geometry of (super)conformal quantum mechanics, Commun. Math. Phys. 213 (2000) 1 [hep-th/9907191] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    S. Cacciatori, D. Klemm and D. Zanon, W algebras, conformal mechanics and black holes, Class. Quant. Grav. 17 (2000) 1731 [hep-th/9910065] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    R. Britto-Pacumio, J. Michelson, A. Strominger and A. Volovich, Lectures on Superconformal Quantum Mechanics and Multi-Black Hole Moduli Spaces, NATO Sci. Ser. C 556 (2000) 255 [hep-th/9911066] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  28. [28]
    G. Papadopoulos, Conformal and superconformal mechanics, Class. Quant. Grav. 17 (2000) 3715 [hep-th/0002007] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    E.E. Donets, A. Pashnev, V.O. Rivelles, D.P. Sorokin and M. Tsulaia, N = 4 superconformal mechanics and the potential structure of AdS spaces, Phys. Lett. B 484 (2000) 337 [hep-th/0004019] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    M.S. Plyushchay, Monopole Chern-Simons term: Charge monopole system as a particle with spin, Nucl. Phys. B 589 (2000) 413 [hep-th/0004032] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    P.K. Ghosh, Conformal symmetry and the nonlinear Schrödinger equation, Phys. Rev. A 65 (2002) 012103 [cond-mat/0102488] [INSPIRE].
  32. [32]
    M. Günaydin, K. Koepsell and H. Nicolai, The Minimal unitary representation of E 8(8), Adv. Theor. Math. Phys. 5 (2002) 923 [hep-th/0109005] [INSPIRE].CrossRefzbMATHGoogle Scholar
  33. [33]
    B. Pioline and A. Waldron, Quantum cosmology and conformal invariance, Phys. Rev. Lett. 90 (2003) 031302 [hep-th/0209044] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    H.E. Camblong and C.R. Ordonez, Anomaly in conformal quantum mechanics: From molecular physics to black holes, Phys. Rev. D 68 (2003) 125013 [hep-th/0303166] [INSPIRE].ADSMathSciNetGoogle Scholar
  35. [35]
    C. Leiva and M.S. Plyushchay, Conformal symmetry of relativistic and nonrelativistic systems and AdS/CFT correspondence, Annals Phys. 307 (2003) 372 [hep-th/0301244] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    B. Pioline and A. Waldron, Automorphic forms: A Physicist’s survey, in Proceedings, Les Houches School of Physics: Frontiers in Number Theory, Physics and Geometry II: On Conformal Field Theories, Discrete Groups and Renormalization, Les Houches, France, March 9–21, 2003, pp. 277–302, (2007) [DOI: https://doi.org/10.1007/978-3-540-30308-4_7] [hep-th/0312068] [INSPIRE].
  37. [37]
    D. Gaiotto, A. Strominger and X. Yin, Superconformal black hole quantum mechanics, JHEP 11 (2005) 017 [hep-th/0412322] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    S. Meljanac and A. Samsarov, Universal properties of conformal quantum many-body systems, Phys. Lett. B 613 (2005) 221 [Erratum ibid. B 620 (2005) 200] [hep-th/0503174] [INSPIRE].
  39. [39]
    C. Duval, G.W. Gibbons and P. Horvathy, Celestial mechanics, conformal structures and gravitational waves, Phys. Rev. D 43 (1991) 3907 [hep-th/0512188] [INSPIRE].ADSMathSciNetGoogle Scholar
  40. [40]
    A. Anabalon, J. Gomis, K. Kamimura and J. Zanelli, N = 4 superconformal mechanics as a non linear realization, JHEP 10 (2006) 068 [hep-th/0607124] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    M. Günaydin, A. Neitzke, B. Pioline and A. Waldron, Quantum Attractor Flows, JHEP 09 (2007) 056 [arXiv:0707.0267] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  42. [42]
    T. Hakobyan and A. Nersessian, Lobachevsky geometry of (super)conformal mechanics, Phys. Lett. A 373 (2009) 1001 [arXiv:0803.1293] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    J. Maldacena, D. Martelli and Y. Tachikawa, Comments on string theory backgrounds with non-relativistic conformal symmetry, JHEP 10 (2008) 072 [arXiv:0807.1100] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    F. Correa, V. Jakubsky and M.S. Plyushchay, Aharonov-Bohm effect on AdS 2 and nonlinear supersymmetry of reflectionless Poschl-Teller system, Annals Phys. 324 (2009) 1078 [arXiv:0809.2854] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  45. [45]
    P.D. Alvarez, J.L. Cortes, P.A. Horvathy and M.S. Plyushchay, Super-extended noncommutative Landau problem and conformal symmetry, JHEP 03 (2009) 034 [arXiv:0901.1021] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  46. [46]
    A. Bagchi and R. Gopakumar, Galilean Conformal Algebras and AdS/CFT, JHEP 07 (2009) 037 [arXiv:0902.1385] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    F. Correa, H. Falomir, V. Jakubsky and M.S. Plyushchay, Hidden superconformal symmetry of spinless Aharonov-Bohm system, J. Phys. A 43 (2010) 075202 [arXiv:0906.4055] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  48. [48]
    T. Hakobyan, S. Krivonos, O. Lechtenfeld and A. Nersessian, Hidden symmetries of integrable conformal mechanical systems, Phys. Lett. A 374 (2010) 801 [arXiv:0908.3290] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    C. Chamon, R. Jackiw, S.-Y. Pi and L. Santos, Conformal quantum mechanics as the CFT 1 dual to AdS 2, Phys. Lett. B 701 (2011) 503 [arXiv:1106.0726] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    Z. Kuznetsova and F. Toppan, D-module Representations of N = 2, 4, 8 Superconformal Algebras and Their Superconformal Mechanics, J. Math. Phys. 53 (2012) 043513 [arXiv:1112.0995] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    S. Fedoruk, E. Ivanov and O. Lechtenfeld, Superconformal Mechanics, J. Phys. A 45 (2012) 173001 [arXiv:1112.1947] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  52. [52]
    K. Andrzejewski, J. Gonera and P. Maślanka, Nonrelativistic conformal groups and their dynamical realizations, Phys. Rev. D 86 (2012) 065009 [arXiv:1204.5950] [INSPIRE].ADSGoogle Scholar
  53. [53]
    J. Gonera, Conformal mechanics, Annals Phys. 335 (2013) 61 [arXiv:1211.4403] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    J. Molina-Vilaplana and G. Sierra, An xp model on AdS 2 spacetime, Nucl. Phys. B 877 (2013) 107 [arXiv:1212.2436] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  55. [55]
    M.S. Plyushchay and A. Wipf, Particle in a self-dual dyon background: hidden free nature and exotic superconformal symmetry, Phys. Rev. D 89 (2014) 045017 [arXiv:1311.2195] [INSPIRE].ADSGoogle Scholar
  56. [56]
    S.J. Brodsky, G.F. de Teramond, H.G. Dosch and J. Erlich, Light-Front Holographic QCD and Emerging Confinement, Phys. Rept. 584 (2015) 1 [arXiv:1407.8131] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    K. Andrzejewski, J. Gonera, P. Kosiński and P. Maślanka, On dynamical realizations of l-conformal Galilei groups, Nucl. Phys. B 876 (2013) 309 [arXiv:1305.6805] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    M. Masuku and J.P. Rodrigues, De Alfaro, Fubini and Furlan from multi Matrix Systems, JHEP 12 (2015) 175 [arXiv:1509.06719] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  59. [59]
    I. Masterov, Remark on higher-derivative mechanics with l-conformal Galilei symmetry, J. Math. Phys. 57 (2016) 092901 [arXiv:1607.02693] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    K. Andrzejewski, Quantum conformal mechanics emerging from unitary representations of SL(2, ℝ), Annals Phys. 367 (2016) 227 [arXiv:1506.05596] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    R. Bonezzi, O. Corradini, E. Latini and A. Waldron, Quantum Mechanics and Hidden Superconformal Symmetry, Phys. Rev. D 96 (2017) 126005 [arXiv:1709.10135] [INSPIRE].ADSMathSciNetGoogle Scholar
  62. [62]
    J. Mateos Guilarte and M.S. Plyushchay, Perfectly invisible \( \mathcal{P}\mathcal{T} \) -symmetric zero-gap systems, conformal field theoretical kinks and exotic nonlinear supersymmetry, JHEP 12 (2017) 061 [arXiv:1710.00356] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    L. Inzunza and M.S. Plyushchay, Hidden superconformal symmetry: Where does it come from?, Phys. Rev. D 97 (2018) 045002 [arXiv:1711.00616] [INSPIRE].ADSMathSciNetGoogle Scholar
  64. [64]
    H. Airault, H.P. McKean and J. Moser, Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem, Commun. Pure Appl. Math. 30 (1977) 95.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  65. [65]
    M. Adler and J. Moser, On a Class of Polynomials Connected with the Korteweg-De Vries Equation, Commun. Math. Phys. 61 (1978) 1 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  66. [66]
    A. Gorsky and N. Nekrasov, Hamiltonian systems of Calogero type and two-dimensional Yang-Mills theory, Nucl. Phys. B 414 (1994) 213 [hep-th/9304047] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  67. [67]
    J.J. Duistermaat and F.A. Grünbaum, Differential equations in the spectral parameter, Commun. Math. Phys. 103 (1986) 177.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  68. [68]
    A.P. Veselov, Huygens’ principle and integrability, Prog. Math. 169 (1998) 259.MathSciNetzbMATHGoogle Scholar
  69. [69]
    S.P. Novikov, S.V. Manakov, L.P. Pitaevskii and V.E. Zakharov, Theory of Solitons, Plenum, New York (1984).zbMATHGoogle Scholar
  70. [70]
    I.M. Krichever, Baker-Akhiezer functions and integrable systems, in Integrability. The Seiberg-Witten and Whitham Equations, H.W. Braden and I.M. Krichever eds., Gordon and Breach Science Publishers, Amsterdam (2000).Google Scholar
  71. [71]
    E.D. Belokolos, A.I. Bobenko, V.Z. Enol’skii, A.R. Its and V.B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer, Berlin (1994).zbMATHGoogle Scholar
  72. [72]
    F. Correa and M.S. Plyushchay, Hidden supersymmetry in quantum bosonic systems, Annals Phys. 322 (2007) 2493 [hep-th/0605104] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  73. [73]
    F. Correa, L.-M. Nieto and M.S. Plyushchay, Hidden nonlinear supersymmetry of finite-gap Lame equation, Phys. Lett. B 644 (2007) 94 [hep-th/0608096] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  74. [74]
    J.L. Burchnall and T.W. Chaundy, Commutative ordinary differential operators, Proc. Lond. Math. Soc. s2-21 (1923) 420 [Proc. Roy. Soc. Lond. A 118 (1928) 557].Google Scholar
  75. [75]
    E.L. Ince, Ordinary differential equations, Dover (1956).Google Scholar
  76. [76]
    F. Correa, V. Jakubsky, L.-M. Nieto and M.S. Plyushchay, Self-isospectrality, special supersymmetry and their effect on the band structure, Phys. Rev. Lett. 101 (2008) 030403 [arXiv:0801.1671] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  77. [77]
    A. Arancibia, J. Mateos Guilarte and M.S. Plyushchay, Effect of scalings and translations on the supersymmetric quantum mechanical structure of soliton systems, Phys. Rev. D 87 (2013) 045009 [arXiv:1210.3666] [INSPIRE].ADSGoogle Scholar
  78. [78]
    C. Leiva and M.S. Plyushchay, Superconformal mechanics and nonlinear supersymmetry, JHEP 10 (2003) 069 [hep-th/0304257] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  79. [79]
    F. Correa, M.A. del Olmo and M.S. Plyushchay, On Hidden broken nonlinear superconformal symmetry of conformal mechanics and nature of double nonlinear superconformal symmetry, Phys. Lett. B 628 (2005) 157 [hep-th/0508223] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  80. [80]
    C.M. Bender, Making sense of non-Hermitian Hamiltonians, Rept. Prog. Phys. 70 (2007) 947 [hep-th/0703096] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  81. [81]
    A. Mostafazadeh, Pseudo-Hermitian Representation of Quantum Mechanics, Int. J. Geom. Meth. Mod. Phys. 7 (2010) 1191 [arXiv:0810.5643] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  82. [82]
    E.P. Wigner, Normal form of antiunitary operators, J. Math. Phys. 1 (1960) 409.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  83. [83]
    A. Arancibia, F. Correa, V. Jakubský, J. Mateos Guilarte and M.S. Plyushchay, Soliton defects in one-gap periodic system and exotic supersymmetry, Phys. Rev. D 90 (2014) 125041 [arXiv:1410.3565] [INSPIRE].ADSGoogle Scholar
  84. [84]
    A. Arancibia and M.S. Plyushchay, Chiral asymmetry in propagation of soliton defects in crystalline backgrounds, Phys. Rev. D 92 (2015) 105009 [arXiv:1507.07060] [INSPIRE].ADSMathSciNetGoogle Scholar
  85. [85]
    C.M. Bender and S. Boettcher, Real spectra in nonHermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80 (1998) 5243 [physics/9712001] [INSPIRE].
  86. [86]
    P. Dorey, C. Dunning and R. Tateo, Spectral equivalences, Bethe Ansatz equations and reality properties in \( \mathcal{P}\mathcal{T} \) -symmetric quantum mechanics, J. Phys. A 34 (2001) 5679 [hep-th/0103051] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  87. [87]
    P. Dorey, C. Dunning and R. Tateo, Supersymmetry and the spontaneous breakdown of \( \mathcal{P}\mathcal{T} \) symmetry, J. Phys. A 34 (2001) L391 [hep-th/0104119] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  88. [88]
    M. Znojil, \( \mathcal{P}\mathcal{T} \) -symmetric harmonic oscillators, Phys. Lett. A 259 (1999) 220 [quant-ph/9905020] [INSPIRE].
  89. [89]
    F. Correa and A. Fring, Regularized degenerate multi-solitons, JHEP 09 (2016) 008 [arXiv:1605.06371] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  90. [90]
    A. Fring and M. Znojil, \( \mathcal{P}\mathcal{T} \) -symmetric deformations of Calogero models, J. Phys. A 41 (2008) 194010 [arXiv:0802.0624] [INSPIRE].ADSzbMATHGoogle Scholar
  91. [91]
    F. Correa and O. Lechtenfeld, \( \mathcal{P}\mathcal{T} \) deformation of angular Calogero models, JHEP 11 (2017) 122 [arXiv:1705.05425] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  92. [92]
    R. El-Ganainy, K.G. Makris, M. Khajavikhan, Z.H. Musslimani, S. Rotter and D.N. Christodoulides, Non-Hermitian physics and PT symmetry, Nat. Phys. 14 (2018) 11.CrossRefGoogle Scholar
  93. [93]
    J.F. Cariñena and M.S. Plyushchay, ABC of ladder operators for rationally extended quantum harmonic oscillator systems, J. Phys. A 50 (2017) 275202 [arXiv:1701.08657] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  94. [94]
    J.F. Cariñena, L. Inzunza and M.S. Plyushchay, Rational deformations of conformal mechanics, Phys. Rev. D 98 (2018) 026017 [arXiv:1707.07357] [INSPIRE].ADSGoogle Scholar
  95. [95]
    A. Arancibia and M.S. Plyushchay, Extended supersymmetry of the self-isospectral crystalline and soliton chains, Phys. Rev. D 85 (2012) 045018 [arXiv:1111.0600] [INSPIRE].ADSGoogle Scholar
  96. [96]
    S.P. Novikov, The periodic problem for the Korteweg-de Vries equation, Funct. Anal. Appl. 8 (1974) 236.MathSciNetCrossRefzbMATHGoogle Scholar
  97. [97]
    L. Inzunza and M.S. Plyushchay, Hidden symmetries of rationally deformed superconformal mechanics, Phys. Rev. D 99 (2019) 025001 [arXiv:1809.08527] [INSPIRE].ADSGoogle Scholar
  98. [98]
    F. Correa, O. Lechtenfeld and M. Plyushchay, Nonlinear supersymmetry in the quantum Calogero model, JHEP 04 (2014) 151 [arXiv:1312.5749] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  99. [99]
    S. Wojciechowski, Superintegrability of the Calogero-Moser system, Phys. Lett. A 95 (1983) 279.ADSMathSciNetCrossRefGoogle Scholar
  100. [100]
    V.B. Kuznetsov, Hidden symmetry of the quantum Calogero-Moser system, Phys. Lett. A 218 (1996) 212 [solv-int/9509001] [INSPIRE].
  101. [101]
    C. Gonera, A note on superintegrability of the quantum Calogero model, Phys. Lett. A 237 (1998) 365.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  102. [102]
    M.F. Rañada, Superintegrability of the Calogero-Moser system: Constants of motion, master symmetries, and time-dependent symmetries, J. Math. Phys. 40 (1999) 236.ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2019

Authors and Affiliations

  1. 1.Departamento de Física Fundamental and IUFFyMUniversidad de SalamancaSalamancaSpain
  2. 2.Departamento de FísicaUniversidad de Santiago de ChileSantiagoChile

Personalised recommendations