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Advances in High Energy Physics
Volume 2017, Article ID 8429863, 13 pages
https://doi.org/10.1155/2017/8429863
Research Article

Exact Solutions of a Class of Double-Well Potentials: Algebraic Bethe Ansatz

Department of Physics, University of Guilan, Rasht 41635-1914, Iran

Correspondence should be addressed to M. Baradaran; moc.oohay@naradarab.eizram

Received 5 September 2017; Revised 15 November 2017; Accepted 19 November 2017; Published 26 December 2017

Academic Editor: Marc de Montigny

Copyright © 2017 M. Baradaran and H. Panahi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

Linked References

  1. G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, “Quantum dynamics of an atomic Bose-Einstein condensate in a double-well potential,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 55, no. 6, pp. 4318–4324, 1997. View at Publisher · View at Google Scholar
  2. M. F. Manning, “Energy levels of a symmetrical double minima problem with applications to the NH3 and ND3 molecules,” The Journal of Chemical Physics, vol. 3, no. 3, pp. 136–138, 1935. View at Publisher · View at Google Scholar
  3. M. Razavy, Quantum Theory of Tunneling, World Scientific Publishing Co. Pte. Ltd., Singapore, 2003. View at Publisher · View at Google Scholar
  4. B. Zhou, J.-Q. Liang, and F.-C. Pud, “Bounces and the calculation of quantum tunneling effects for the asymmetric double-well potential,” Physics Letters A, vol. 271, no. 1-2, pp. 26–30, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  5. A. Heuer and R. J. Silbey, “Microscopic description of tunneling systems in a structural model glass,” Physical Review Letters, vol. 70, no. 25, pp. 3911–3914, 1993. View at Publisher · View at Google Scholar · View at Scopus
  6. T. Wu, “Characteristic Values of the two minima problem and quantum defects of fstates of heavy atoms,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 44, no. 9, pp. 727–731, 1933. View at Publisher · View at Google Scholar
  7. C. M. Bender and G. V. Dunne, “Quasi-exactly solvable systems and orthogonal polynomials,” Journal of Mathematical Physics, vol. 37, no. 1, pp. 6–11, 1996. View at Publisher · View at Google Scholar
  8. M. Razavy, “An exactly soluble Schrödinger equation with a bistable potential,” American Journal of Physics, vol. 48, no. 4, article 285, 1980. View at Publisher · View at Google Scholar
  9. M. A. Shifman, “New findings in quantum mechanics (partial algebraization of the spectral problem),” International Journal of Modern Physics A, vol. 4, no. 12, pp. 2897–2952, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  10. N. Kamran and P. J. Olver, “Lie algebras of differential operators and Lie-algebraic potentials,” Journal of Mathematical Analysis and Applications, vol. 145, no. 2, pp. 342–356, 1990. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. A. González-López, N. Kamran, and P. J. Olver, “Normalizability of one-dimensional quasi-exactly solvable Schrödinger operators,” Communications in Mathematical Physics, vol. 153, no. 1, pp. 117–146, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  12. A. V. Turbiner, “Lie-algebras and linear operators with invariant subspaces,” Contemporary Mathematics, vol. 160, pp. 263–310, 1994. View at Google Scholar
  13. B. Bagchi, C. Quesne, and R. Roychoudhury, “A complex periodic QES potential and exceptional points,” Journal of Physics A: Mathematical and General, vol. 41, no. 2, Article ID 022001, 2008. View at Google Scholar · View at MathSciNet
  14. A. G. Ushveridze, Quasi-Exactly Solvable Models in Quantum Mechanics, IOP, Bristol, UK, 1994. View at MathSciNet
  15. C.-M. Chiang and C.-L. Ho, “Charged particles in external fields as physical examples of quasi-exactly-solvable models: a unified treatment,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 63, no. 6, Article ID 062105, 2001. View at Publisher · View at Google Scholar
  16. C.-L. Ho, “Quasi-exact solvability of Dirac equation with Lorentz scalar potential,” Annals of Physics, vol. 321, no. 9, pp. 2170–2182, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  17. C.-L. Ho, “Prepotential approach to exact and quasi-exact solvabilities,” Annals of Physics, vol. 323, no. 9, pp. 2241–2252, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  18. C.-L. Ho and P. Roy, “Quasi-exact solvability of DIRac-Pauli equation and generalized DIRac oscillators,” Annals of Physics, vol. 312, no. 1, pp. 161–176, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  19. Y.-Z. Zhang, “Exact polynomial solutions of second order differential equations and their applications,” Journal of Physics A: Mathematical and Theoretical, vol. 45, no. 6, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  20. S. Negro, “Integrable structures in quantum field theory,” Journal of Physics A: Mathematical and General, vol. 49, no. 32, Article ID 323006, 2016. View at Google Scholar · View at MathSciNet
  21. N. Y. Reshetikhin and P. B. Wiegmann, “Towards the classification of completely integrable quantum field theories (the Bethe-ansatz associated with Dynkin diagrams and their automorphisms),” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 189, no. 1-2, pp. 125–131, 1987. View at Google Scholar · View at MathSciNet
  22. P. T. Fernandes, P. C. B. Salgado, A. L. A. Noronha, F. D. Barbosa, E. A. P. Souza, and L. M. Li, “Stigma scale of epilepsy: conceptual issues,” Journal of Epilepsy and Clinical Neurophysiology, vol. 10, no. 4, pp. 213–218, 2004. View at Google Scholar · View at Scopus
  23. A. Doikou, P. P. Martin, and J. Stat, “On quantum group symmetry and Bethe ansatz for the asymmetric twin spin chain with integrable boundary,” Journal of Statistical Mechanics: Theory and Experiment, vol. 6, Article ID P06004, 2006. View at Google Scholar
  24. K. Sogo, “Explicit construction of the second-quantized anyon operators,” Journal of the Physical Society of Japan, vol. 64, no. 7, pp. 2249–2251, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  25. H. Tütüncüler, R. Koç, and E. Olğar, “Solution of a Hamiltonian of quantum dots with Rashba spin-orbit coupling: quasi-exact solution,” Journal of Physics A: Mathematical and General, vol. 37, no. 47, pp. 11431–11438, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  26. G. Arutyunov, S. Frolov, and M. Staudacher, “Bethe ansatz for quantum strings,” Journal of High Energy Physics, vol. 2004, no. 10, 16 pages, 2004. View at Publisher · View at Google Scholar
  27. M. Salazar-Ramírez, D. Ojeda-Guillén, and R. D. Mota, “Algebraic approach and coherent states for a relativistic quantum particle in cosmic string spacetime,” Annals of Physics, vol. 372, pp. 283–296, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  28. Ö. Yeşiltaş, “ solutions for the relativistic quantum particle in cosmic string spacetime,” The European Physical Journal Plus, vol. 130, article 128, 2015. View at Publisher · View at Google Scholar
  29. E. A. Bergshoeff, J. Hartong, M. Hübscher, and T. Ortín, “Stringy cosmic strings in matter coupled N = 2, d = 4 supergravity,” Journal of High Energy Physics, vol. 2008, no. 5, 33 pages, 2008. View at Publisher · View at Google Scholar
  30. I. H. Russell and D. J. Toms, “Symmetry breaking around cosmic strings,” Classical and Quantum Gravity, vol. 6, no. 10, pp. 1343–1349, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  31. S. Zarrinkamar, H. Hassanabadi, and A. A. Rajabi, “What is the most simple solution of Wheeler-DeWitt equation?” Astrophysics and Space Science, vol. 343, no. 1, pp. 391–393, 2013. View at Publisher · View at Google Scholar · View at Scopus
  32. H. Panahi, S. Zarrinkamar, and M. Baradaran, “The wheeler-DeWitt equation in filćhenkov model: the lie algebraic approach,” Zeitschrift fur Naturforschung, vol. 71, no. 11, pp. 1021–1026, 2016. View at Publisher · View at Google Scholar · View at Scopus
  33. C. S. Park, M. G. Jeong, S. Yoo, and D. K. Park, “Double-well potential: The WKB approximation with phase loss and anharmonicity effect,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 58, no. 5, pp. 3443–3447, 1998. View at Publisher · View at Google Scholar
  34. V. S. Shchesnovich and M. Trippenbach, “Fock-space WKB method for the boson Josephson model describing a Bose-Einstein condensate trapped in a double-well potential,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 78, no. 2, 2008. View at Publisher · View at Google Scholar
  35. A. J. Sous, “Eigenenergies for the Razavy potential using the asymptotic iteration method,” Modern Physics Letters A, vol. 22, no. 22, pp. 1677–1684, 2007. View at Google Scholar · View at MathSciNet
  36. Q.-T. Xie, “New quasi-exactly solvable double-well potentials,” Journal of Physics A: Mathematical and General, vol. 45, no. 17, Article ID 175302, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  37. J.-Q. Liang and H. J. W. Müller-Kirsten, “Periodic instantons and quantum-mechanical tunneling at high energy,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 46, no. 10, pp. 4685–4690, 1992. View at Publisher · View at Google Scholar · View at Scopus
  38. J.-Q. Liang and H. J. M\"uller-Kirsten, “Quantum tunneling for the sine-Gordon potential: energy band structure and Bogomolny-Fateyev relation,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 51, no. 2, pp. 718–725, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  39. Y. Jin, Y. Nie, J. Liang, Z. Chen, W. Xie, and F. Pu, “Tunnel splitting in biaxial spin particles as a function of applied magnetic field,” Physical Review B: Condensed Matter and Materials Physics, vol. 62, no. 5, pp. 3316–3321, 2000. View at Publisher · View at Google Scholar
  40. J. Liang and H. Müller-Kirsten, “Quantum mechanical tunnelling at finite energy and its equivalent amplitudes in the (vacuum) instanton approximation,” Physics Letters B, vol. 332, no. 1-2, pp. 129–135, 1994. View at Publisher · View at Google Scholar
  41. F. Finkel, A. González-López, and M. A. Rodríguez, “On the families of orthogonal polynomials associated to the Razavy potential,” Journal of Physics A: Mathematical and General, vol. 32, no. 39, pp. 6821–6835, 1999. View at Publisher · View at Google Scholar · View at Scopus
  42. M. C. Lawrence and G. N. Robertson, “Estimating the proton potential in KDP from infrared and crystallographic data,” Ferroelectrics, vol. 34, no. 1, pp. 179–186, 1981. View at Publisher · View at Google Scholar · View at Scopus
  43. X. Duan and S. Scheiner, “Analytic functions fit to proton transfer potentials,” Journal of Molecular Structure, vol. 270, pp. 173–185, 1992. View at Publisher · View at Google Scholar · View at Scopus
  44. S. Habib, A. Khare, and A. Saxena, “Statistical mechanics of double sinh-Gordon kinks,” Physica D: Nonlinear Phenomena, vol. 123, no. 1-4, pp. 341–356, 1998. View at Publisher · View at Google Scholar · View at Scopus