Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


Integrability of a Coupled Harmonic Oscillator in Extended Complex Phase Space

Discontinuity, Nonlinearity, and Complexity 4(1) (2016) 35--48 | DOI:10.5890/DNC.2016.03.004

Ram Mehar Singh

Department of Physics, Ch. Devi Lal University, Sirsa-125055, Haryana, India

Download Full Text PDF

 

Abstract

With in the frame work of extended complex phase space characterized by x = x1 + ip3, y = x2 + ip4, px = p1 + ix3 and py = p2 + ix4, we investigate the exact invariants for a coupled harmonic oscillator along with PT-symmetric version in two dimensions. For this purpose rationalization method is employed and the invariants obtained in this work play an important role to study the complex trajectories of the concerned classical system.

Acknowledgments

The author expresses his gratitude to Prof. S.C. Mishra and Dr. Fakir Chand, Department of Physics, Kurukshetra University, Kurukshetra (India), for their valuable suggestions regarding the manuscript. He is also thankful to the referees for their useful comments which helped in fine-tuning of some basic ideas in original version of the paper.

References

  1. [1]  Kaushal, R.S.(1998),Classical and Quantum Mechanics of Noncentral Potentials, Narosa Publishing House, New Delhi.
  2. [2]  Moiseyev, N.(1998), Quantum theories of resonances: Calculating energies width and cross-sections by complex scaling, Physics Reports, 302, 211-293.
  3. [3]  Colgerave, R.K., Croxson, P., and Mannan, M.A.,(1988), Complex invariants for the time dependent harmonic oscillator, Physics Letters , A131, 407-410.
  4. [4]  Kaushal, R.S. and Singh, S.(2001), Construction of complex invariants for classical dynamical systems, Annals of Physics (N.Y.), 288, 253-276.
  5. [5]  Nelson, D.R. and Snerb, N.M.(1998), Non-hermitian localization and population biology, Physical Review , E58(2), 1383-1403.
  6. [6]  Hollowood, T.J.(1992), Solitons in affine toda theories, Nuclear Physics, B384, 523-540.
  7. [7]  Korsch, H.J.(1982), On Classsical and Quantum integrability, Physics Letters A, 90, 113-114.
  8. [8]  Hatano, N. and Nelson, D.R.(1996), Localization transition in PT-symmetric quantum mechanics, Physical Review Letter, 77, 570-573;(1997)Vortex pinning and non-hermitian quantum mechanics, Physical Review B, 56, 8651-8673.
  9. [9]  Verheest, F.(1987), Nonlinear wave interaction in a complex Hamiltonian formalism, Journal of Physics A: Mathematical and General, 20, 103-110.
  10. [10]  Bender,C.M. and Boettcher, S.(1998), Real spectra in non-hermitian Hamiltonians having PT-symmetry, Physical Review Letter, 80, 5243-5246.
  11. [11]  Bender, C.M., Brody, D.C., and Jones, H.F.(2003), Must a Hamiltonian be hermitian, American Journal of Physics, 71,1095-1102.
  12. [12]  Bender, C.M., Brod, J., Refig, A., and Reuter, M.E. (2004), The C operator in PT-quantum theories , Journal of Physics A, 37, 10139-10165.
  13. [13]  Kaushal, R.S. and Parthasarthi(2002),Quantum mechanics of complex Hamiltonian systems in one dimension, Journal of Physics A, 35, 8743-8761.
  14. [14]  Parthasarthi and Kaushal, R.S.(2003), Quantum mechanics of complex sextic potential in one dimension, Physica Scripta, 68, 115-127.
  15. [15]  Xavier Jr., A.L. and de Aguiar, M.A.M.(1996), Complex trajectories in the quartic oscillator and its semiclassical coherent state, Annals of Physics, 252, 458-476; (1997)Phase space approach to the tunnel effect: A new semiclassical traverse time, Physical Review Letter, 79, 3323-3326.
  16. [16]  Goldstien, H. (1981), Classical Mechanics, AddisionWisley.
  17. [17]  Kaushal, R.S., Mishra,S.C., and Tripathy, K.C. (1985), Construction of Second Constant of Motion for Two Dimensional Classical Systems, Journal of Mathematical Physics, 26(3),420-427.
  18. [18]  Hietarinta, J.(1983), A search for integrable two-dimensionalHamiltonian systems with polynomial potential, Physical Review, A96, 273-278.
  19. [19]  Hall, L.S.(1985), Invariant polynomial in momenta for integrable Hamiltonians, Physical Review Letter, 54, 614-615.
  20. [20]  Tabor, M. (1989), Chaos and Integrability in Nonlinear Dynamics, Wiley, New York.
  21. [21]  Chand, F. , Singh, R.M., Kumar, N., and Mishra, S.C. (2007), The solution of the Schrödinger equation for complex Hamiltonian systems in two dimensions, Journal of Physics A: Mathematical and Theoretical, 40, 10171-10182.