Papers
Topics
Authors
Recent
Search
2000 character limit reached

Non-stationary SQM/IST Correspondence and ${\cal CPT}/{\cal PT}$-invariant paired Hamiltonians on the line

Published 29 Feb 2024 in hep-th and quant-ph | (2402.19066v3)

Abstract: We fill some of existed gaps in the correspondence between Supersymmetric Quantum Mechanics and the Inverse Scattering Transform by extending the consideration to the case of paired stationary and non-stationary Hamiltonians. We formulate the corresponding to the case Goursat problem and explicitly construct the kernel of the non-local Inverse Scattering Transform, which solves it. As a result, we find the way of constructing non-hermitian Hamiltonians from the initially hermitian ones, that leads, in the case of real-valued spectra of both potentials, to pairing of ${\cal CPT/PT}$-invariant Hamiltonians. The relevance of our proposal to Quantum Optics and optical waveguides technology, as well as to non-linear dynamics and Black Hole Physics is briefly discussed.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (85)
  1. G. L. Lamb, “ELEMENTS OF SOLITON THEORY,” Wiley&Sons, 1980, ISBN 0-471-04559-4
  2. W. Eckhaus and A. Van Harten, “THE INVERSE SCATTERING TRANSFORMATION AND THE THEORY OF SOLITONS. AN INTRODUCTION,” North Holland, 1981, ISBN 0-444-86166-1
  3. P. G. Drazin and R. S. Johnson, “Solitons: An Introduction,” Cambridge Univ. Press, 1989,
  4. Peter J. Olver, David H. Sattinger (Eds.), “Solitons in Physics, Mathematics, and Nonlinear Optics,” The IMA Volumes in Mathematics and Its Applications 25, Springer, New York, 1990, ISBN 978-1-4613-9035-0
  5. B. N. Zakhariev and A. A. Suzko, “Direct and Inverse Problems,” Springer, Berlin, Heidelberg, 1990, ISBN 978-3-540-52484-7 doi:10.1007/978-3-642-95615-7
  6. H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers and E. Herlt, “Exact solutions of Einstein’s field equations,” Cambridge Univ. Press, 2003, ISBN 978-0-521-46702-5, 978-0-511-05917-9 doi:10.1017/CBO9780511535185
  7. A. Krasinski, “Inhomogeneous cosmological models,” Cambridge Univ. Press, 2011, ISBN 978-0-511-88754-3, 978-0-521-48180-9, 978-0-521-03017-5
  8. V. Bargmann, “On the Connection between Phase Shifts and Scattering Potential,” Rev. Mod. Phys. 21, 488 (1949) doi:10.1103/RevModPhys.21.488
  9. I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation from its spectral function” Izv. Akad. Nauk SSSR Ser. Mat. 15, 309-360 (1951). (in Russian) [Gel’fand, I. M.; Levitan, B. M. On the determination of a differential equation from its spectral function.Amer. Math. Soc. Transl. 1, 253-304 (1955). (English translation)]
  10. M. G. Krein, “On the method of effective solution of an inverse boundary value problem,” Dokl. Akad. Nauk SSSR 94 6 (1954). (in Russian)
  11. V. A. Marchenko, “Reconstruction of the potential energy from the phases of scattered waves,” Dokl. Akad. Nauk SSSR 104 5 (1955). (in Russian)
  12. L. D. Faddeev, “The Inverse Problem in the Quantum Theory of Scattering,” J. Math. Phys. 4, 72-104 (1963).
  13. N. J. Zabusky and M. D. Kruskal, “Interaction of ’Solitons’ in a Collisionless Plasma and the Recurrence of Initial States,” Phys. Rev. Lett. 15, 240-243 (1965) doi:10.1103/PhysRevLett.15.240
  14. C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, “Method for solving the Korteweg-deVries equation,” Phys. Rev. Lett. 19, 1095-1097 (1967) doi:10.1103/PhysRevLett.19.1095
  15. M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, “The Inverse scattering transform fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249-315 (1974)
  16. E. Witten, “Dynamical Breaking of Supersymmetry,” Nucl. Phys. B 188, 513 (1981) doi:10.1016/0550-3213(81)90006-7
  17. F. Cooper and B. Freedman, “Aspects of Supersymmetric Quantum Mechanics,” Annals Phys. 146, 262 (1983) doi:10.1016/0003-4916(83)90034-9
  18. M. de Crombrugghe and V. Rittenberg, “Supersymmetric Quantum Mechanics,” Annals Phys. 151, 99 (1983) doi:10.1016/0003-4916(83)90316-0
  19. C. M. Bender, F. Cooper and B. Freedman, “A New Strong Coupling Expansion for Quantum Field Theory Based on the Langevin Equation,” Nucl. Phys. B 219, 61-80 (1983) doi:10.1016/0550-3213(83)90428-5
  20. E. Gozzi, “Ground-state wave-function ”representation”,” Phys. Lett. B 129, 432-436 (1983) [erratum: Phys. Lett. B 134, 477-477 (1984)] doi:10.1016/0370-2693(83)90134-X
  21. A. Comtet, A. D. Bandrauk and D. K. Campbell, “EXACTNESS OF SEMICLASSICAL BOUND STATE ENERGIES FOR SUPERSYMMETRIC QUANTUM MECHANICS,” Phys. Lett. B 150, 159-162 (1985) doi:10.1016/0370-2693(85)90160-1
  22. P. Kumar, M. Ruiz-Altaba and B. S. Thomas, “Tunneling Exchange, Supersymmetry and the Riccati Equation,” Phys. Rev. Lett. 57, 2749-2751 (1986) doi:10.1103/PhysRevLett.57.2749
  23. Akira Hasegawa and Masayuki Matsumoto, “Optical Solitons in Fibers,” Springer-Verlag Berlin Heidelberg New York, 2003 (3rd Edition), ISBN 978-3-642-07826-2 doi:10.1007/978-3-540-46064-0
  24. Linn F. Mollenauer and James P. Gordon, “Solitons in optical fibers: fundamentals and applications,” Elsevier Academic Press, 2006, ISBN-13: 978-0-12-504190-4
  25. Supriyo Datta, “Quantum Transport: Atom to Transistor,” Cambridge University Press, 2005, ISBN: 9780521631457,0521631459
  26. Takaaki Tsurumi, Hiroyuki Hirayama, Martin Vacha, Tomoyasu Taniyama, “Nanoscale Physics for Materials Science,” CRC Press, 2009, ISBN: 978-1-4398-0059-1
  27. S. Ward, R. Allahverdi and A. Mafi, “Supersymmetric Analysis of Stochastic Micro-Bending in Optical Waveguides,” doi:10.48550/arXiv.2009.11847 [arXiv:2009.11847 [physics.optics]].
  28. V. P. Berezovoj and M. I. Konchatnij, “Dynamics of localized states in extended supersymmetric quantum mechanics with multi-well potentials,” J. Phys. A 45, 225302 (2012) doi:10.1088/1751-8113/45/22/225302 [arXiv:1107.2523 [hep-th]].
  29. V. P. Berezovoj, M. I. Konchatnij and A. J. Nurmagambetov, “Tunneling dynamics in exactly-solvable models with triple-well potentials,” J. Phys. A 46, 065302 (2013) doi:10.1088/1751-8113/46/6/065302 [arXiv:1209.0752 [quant-ph]].
  30. A. Macho, R. Llorente and C. Garcίa-Meca, “Supersymmetric Transformations in Optical Fibers,” Phys. Rev. Applied 9, no.1, 014024 (2018) doi:10.1103/PhysRevApplied.9.014024
  31. V. P. Berezovoj, M. I. Konchatnij and A. J. Nurmagambetov, “Hallmarks of tunneling dynamics with broken reflective symmetry,” Nucl. Phys. B 969, 115483 (2021) doi:10.1016/j.nuclphysb.2021.115483 [arXiv:2012.11888 [quant-ph]].
  32. H. E. Moses, “A Generalization of the Gelfand-Levitan Equation for the One-Dimensional Schrödinger Equation,” J. Math. Phys. 18, 2243-2250 (1977).
  33. B. Defacio and H. E. Moses, “THE GELFAND-LEVITAN EQUATION CAN GIVE SIMPLE EXAMPLES OF NONSELFADJOINT OPERATORS WITH COMPLETE EIGENFUNCTIONS AND SPECTRAL REPRESENTATIONS. 1. GHOSTS AND RESONANCES,” J. Math. Phys. 21, 1716-1723 (1980) doi:10.1063/1.524619
  34. P.  A. Deift, (1978), “Applications of a commutation formula,” Duke Math. J. 45, 267-310 (1978) doi:10.1215/s0012-7094-78-04516-7
  35. P. Deift and E. Trubowitz, “Inverse scattering on the line,” Comm. Pure Appl. Math. 32, 121-251 (1979) doi:10.1002/cpa.3160320202
  36. P. B. Abraham and H. E. Moses, “Changes in potentials due to changes in the point spectrum: Anharmonic oscillator with exact solutions,” Phys. Rev. A 22, 1333-1340 (1980) doi:10.1103/PhysRevA.22.1333
  37. M. M. Nieto, “Relationship Between Supersymmetry and the Inverse Method in Quantum Mechanics,” Phys. Lett. B 145, 208-210 (1984) doi:10.1016/0370-2693(84)90339-3
  38. C. V. Sukumar,“Supersymmetric quantum mechanics and the inverse scattering method,” J. Phys. A 18, 2937-2955 (1985) doi:10.1088/0305-4470/18/15/021
  39. W. Kwong and J. L. Rosner, “Supersymmetric Quantum Mechanics and Inverse Scattering,” Prog. Theor. Phys. Suppl. 86, 366 (1986) doi:10.1143/PTPS.86.366
  40. D. L. Pursey, “New families of isospectral Hamiltonians,” Phys. Rev. D 33, 1048-1055 (1986) doi:10.1103/PhysRevD.33.1048
  41. M. Luban and D. L. Pursey, “New Schrodinger equations for old: Inequivalence of the Darboux and Abraham-Moses constructions,” Phys. Rev. D 33, 431-436 (1986) doi:10.1103/PhysRevD.33.431
  42. V. P. Berezovoi and A. I. Pashnev, “N=2𝑁2N=2italic_N = 2 Supersymmetric Quantum Mechanics and the Inverse Scattering Problem,” Theor. Math. Phys. 74, 264-268 (1988) doi:10.1007/BF01016619
  43. C. V. Sukumar, “Supersymmetric transformations and Hamiltonians generated by the Marchenko equations,” J. Phys. A 21, L455-L458 (1988) doi:10.1088/0305-4470/21/8/005
  44. A. Khare and U. Sukhatme, “Phase Equivalent Potentials Obtained From Supersymmetry,” J. Phys. A 22, 2847 (1989) doi:10.1088/0305-4470/22/14/031
  45. V. P. Berezovoj and A. I. Pashnev, “Extended N=2 supersymmetric quantum mechanics and isospectral Hamiltonians,” Z. Phys. C 51, 525-529 (1991) doi:10.1007/BF01548580
  46. S. Kabanikhin, M. Shishlenin, N. Novikov, N. Prokhoshin, “Spectral, Scattering and Dynamics: Gelfand–Levitan–Marchenko–Krein Equations,” Mathematics 11 4458 (2023) doi:10.3390/math11214458
  47. B. Mielnik, “Factorization method and new potentials with the oscillator spectrum,” J. Math. Phys. 25, 3387-9 (1984) doi:10.1063/1.526108
  48. N. G. van Kampen, “A soluble model for diffusion in a bistable potential,” J. Stat. Phys. 17, 71-88 (1977) doi:10.1007/BF01268919
  49. V. G. Bagrov, I. I. Ovcharov and B. F. Samsonov, “Playing with one dimensional Schrodinger equation,” Trudy FORA 1, 1-15 (1996). (In Russian)
  50. B. Mielnik and O. Rosas-Ortiz, “Factorization: Little or great algorithm?,” J. Phys. A 37, 10007-10035 (2004) doi:10.1088/0305-4470/37/43/001
  51. V. G. Bagrov, A. V. Shapovalov and I. V. Shirokov, “A new method of exact solution generation for the one-dimensional Schrödinger equation,” Phys. Lett. A 147 348-350 (1990)
  52. V. G. Bagrov, A. V. Shapovalov and I. V. Shirokov, “Generation of new exactly solvable potentials of a nonstationary Schrödinger equation,” Theoretical and Mathematical Physics 87 635-640 (1991) doi:10.1007/bf01017951
  53. V. G. Bagrov, B. F. Samsonov and L. A Shekoyan, “Darboux transformations for the nonsteady Schrodinger equation,” Russian Phys. Jour. 38 706-712 (1995) doi:10.1007/BF00560273
  54. V. G. Bagrov and B. F. Samsonov, “Coherent states for anharmonic oscillator Hamiltonians with equidistant and quasi-equidistant spectra,” J. Phys. A 29 1011-1023 (1996) doi:10.1088/0305-4470/29/5/015
  55. V. G. Bagrov and B. F. Samsonov, “Supersymmetry of a nonstationary Schrödinger equation,” Phys. Lett. A 210 60-64 (1996) doi:10.1016/0375-9601(95)00832-2
  56. F. Cannata, M. V. Ioffe, G. Junker and D. Nishnianidze, “Intertwining relations of nonstationary Schrödinger operators,” J. Phys. A 32, 3583-3598 (1999) doi:10.1088/0305-4470/32/19/309 [arXiv:quant-ph/9810033 [quant-ph]].
  57. A. Schulze-Halberg, E. Pozdeeva and A. Suzko, “Explicit Darboux transformations of arbitrary order for generalized time-dependent Schrödinger equations,” Journal of Physics A: Mathematical and Theoretical 42, no.11, 115211 (2009) doi:10.1088/1751-8113/42/11/115211
  58. A. A. Suzko and A. Schulze-Halberg, “Darboux transformations and supersymmetry for the generalized Schroedinger equations in (1+1) dimensions,” J. Phys. A 42, 295203 (2009) doi:10.1088/1751-8113/42/29/295203
  59. K. Zelaya and O. Rosas-Ortiz, “Exactly Solvable Time-Dependent Oscillator-Like Potentials Generated by Darboux Transformations,” J. Phys. Conf. Ser. 839, no.1, 012018 (2017) doi:10.1088/1742-6596/839/1/012018 [arXiv:1706.04697 [quant-ph]].
  60. D. Rasinskaitė and P. Strange, “Quantum surfing,” Eur. J. Phys. 42, no.1, 015402 (2020) doi:10.1088/1361-6404/abbab0
  61. P. Strange, “Quantum potential in time-dependent supersymmetric quantum mechanics,” Phys. Rev. A 104, no.6, 062213 (2021) doi:10.1103/PhysRevA.104.062213
  62. D. Baye, G. Lévai and J. M. Sparenberg, “Phase-equivalent complex potentials,” Nucl. Phys. A 599, 435-456 (1996) doi:10.1016/0375-9474(95)00487-4
  63. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243-5246 (1998) doi:10.1103/PhysRevLett.80.5243 [arXiv:physics/9712001 [physics]].
  64. C. M. Bender, S. Boettcher and P. Meisinger, “PT symmetric quantum mechanics,” J. Math. Phys. 40, 2201-2229 (1999) doi:10.1063/1.532860 [arXiv:quant-ph/9809072 [quant-ph]].
  65. F. Cannata, G. Junker and J. Trost, “Schrödinger operators with complex potential but real spectrum,” Phys. Lett. A 246, 219-226 (1998) doi:10.1016/S0375-9601(98)00517-9 [arXiv:quant-ph/9805085 [quant-ph]].
  66. P. Dorey, C. Dunning and R. Tateo, “Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics,” J. Phys. A 34, 5679-5704 (2001) doi:10.1088/0305-4470/34/28/305 [arXiv:hep-th/0103051 [hep-th]].
  67. P. Dorey, C. Dunning and R. Tateo, “Supersymmetry and the spontaneous breakdown of PT symmetry,” J. Phys. A 34, L391 (2001) doi:10.1088/0305-4470/34/28/102 [arXiv:hep-th/0104119 [hep-th]].
  68. M. Znojil, F. Cannata, B. Bagchi and R. Roychoudhury, “Supersymmetry without hermiticity within PT symmetric quantum mechanics,” Phys. Lett. B 483, 284-289 (2000) doi:10.1016/S0370-2693(00)00569-4 [arXiv:hep-th/0003277 [hep-th]].
  69. C. M. Bender, D. C. Brody and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002) [erratum: Phys. Rev. Lett. 92, 119902 (2004)] doi:10.1103/PhysRevLett.89.270401 [arXiv:quant-ph/0208076 [quant-ph]].
  70. Z. Ahmed, C. M. Bender and M. V. Berry, “Reflectionless Potentials and PT Symmetry,” J. Phys. A 38, L627-L630 (2005) doi:10.1088/0305-4470/38/39/L01 [arXiv:quant-ph/0508117 [quant-ph]].
  71. F. Cannata, J. P. Dedonder and A. Ventura, “Scattering in PT symmetric quantum mechanics,” Annals Phys. 322, 397-433 (2007) doi:10.1016/j.aop.2006.05.011 [arXiv:quant-ph/0606129 [quant-ph]].
  72. O. Rosas-Ortiz, O. Castaños and D. Schuch, “New supersymmetry-generated complex potentials with real spectra,” J. Phys. A 48, no.44, 445302 (2015) doi:10.1088/1751-8113/48/44/445302 [arXiv:1505.05197 [quant-ph]].
  73. J. Cen, A. Fring and T. Frith, “Time-dependent Darboux (supersymmetric) transformations for non-Hermitian quantum systems,” J. Phys. A 52, no.11, 115302 (2019) doi:10.1088/1751-8121/ab0335 [arXiv:1811.00149 [quant-ph]].
  74. C. M. Bender, P. E. Dorey, C. Dunning, A. Fring, D. W. Hook, H. F. Jones, S. Kuzhel, G. Lévai and R. Tateo, “PT Symmetry,” WSP, 2019, doi:10.1142/q0178
  75. T. Frith, “Time-dependence in non-Hermitian quantum systems,” [arXiv:2002.01977 [quant-ph]].
  76. C. M. Bender and D. W. Hook, “PT-symmetric quantum mechanics,” [arXiv:2312.17386 [quant-ph]].
  77. P. Pereshogin and P. Pronin, “Effective Hamiltonian and Berry phase in a quantum mechanical system with time dependent boundary conditions,” Phys. Lett . A 156 12-16 (1991) doi:10.1016/0375-9601(91)90117-Q
  78. A. Koller and M. Olshanii, “Supersymmetric Quantum Mechanics and Solitons of the sine-Gordon and Nonlinear Schrödinger Equations,” Phys. Rev. E 84, 066601 (2011) doi:10.1103/PhysRevE.84.066601 [arXiv:1012.2843 [math-ph]].
  79. P. G. Kevrekidis, J. Cuevas–Maraver, A. Saxena, F. Cooper and A. Khare, “Interplay between parity-time symmetry, supersymmetry, and nonlinearity: An analytically tractable case example,” Phys. Rev. E 92, 042901 (2015) doi:10.1103/PhysRevE.92.042901
  80. B. Bagchi, R. Ghosh and S. Sen, “Analogue Hawking Radiation as a Tunneling in a Two-Level -Symmetric System,” Entropy 25, no.8, 1202 (2023) doi:10.3390/e25081202 [arXiv:2304.14174 [gr-qc]].
  81. V. P. Berezovoj, G. I. Ivashkevych and M. I. Konchatnij, “Exactly solvable diffusion models in the framework of the extended supersymmetric quantum mechanics,” Phys. Lett. A 374, 1197-1200 (2010) doi:10.1016/j.physleta.2009.12.066
  82. G. R. Peglow Borges, E. Drigo Filho and R. M. Ricotta, “Variational supersymmetric approach to evaluate Fokker-Planck probability,” Physica A 389, 3892-3899 (2010) doi:10.1016/j.physa.2010.05.027
  83. D. J. Fernandez C, “Trends in supersymmetric quantum mechanics,” doi:10.1007/978-3-030-20087-9_2 [arXiv:1811.06449 [quant-ph]].
  84. K. Glampedakis, A. D. Johnson and D. Kennefick, “Darboux transformation in black hole perturbation theory,” Phys. Rev. D 96, no.2, 024036 (2017) doi:10.1103/PhysRevD.96.024036 [arXiv:1702.06459 [gr-qc]].
  85. D. Li, A. Hussain, P. Wagle, Y. Chen, N. Yunes and A. Zimmerman, “Isospectrality breaking in the Teukolsky formalism,” [arXiv:2310.06033 [gr-qc]].

Summary

No one has generated a summary of this paper yet.

Sign Up to Summarize

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Generate Now

Collections

Sign up for free to add this paper to one or more collections.

Sign Up

Tweets

Sign up for free to view the 3 tweets with 1 like about this paper.

Sign Up for Free