Nuclear Physics and Atomic Energy

ядерна ф≥зика та енергетика
Nuclear Physics and Atomic Energy

  ISSN: 1818-331X (Print), 2074-0565 (Online)
  Publisher: Institute for Nuclear Research of the National Academy of Sciences of Ukraine
  Languages: Ukrainian, English
  Periodicity: 4 times per year

  Open access peer reviewed journal


 Home page   About 
Nucl. Phys. At. Energy 2021, volume 22, issue 2, pages 127-142.
Section: Nuclear Physics.
Received: 04.11.2020; Accepted: 02.04.2021; Published online: 10.09.2021.
PDF Full text (ua)
https://doi.org/10.15407/jnpae2021.02.127

On the quantum anharmonic oscillator and Padé approximations

V. A. Babenko*, N. M. Petrov

Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

*Corresponding author. E-mail address: pet2@ukr.net

Abstract: For the quantum quartic anharmonic oscillator with the Hamiltonian H = (p2+x2)/2+λx4, which is one of the traditional quantum-mechanical and quantum-field-theory models, we study summation of its factorially divergent perturbation series by the proposed method of averaging of the corresponding Padé approximants. Thus, for the first time, we are able to construct the Padé-type approximations that possess correct asymptotic behaviour at infinity with a rise of the coupling constant λ. The approach gives very essential theoretical and applicatory-computational advantages in applications of the given method. We also study convergence of the applied approximations and calculate by the proposed method the ground state energy E0(λ) of the anharmonic oscillator for a wide range of variation of the coupling constant λ.

Keywords: anharmonic oscillator, quantum field theory, perturbation theory, Padé approximants.

References:

1. C.M. Bender, T.T. Wu. Anharmonic oscillator. Phys. Rev. 184 (1969) 1231. https://doi.org/10.1103/PhysRev.184.1231

2. D.I. Kazakov, D.V. Shirkov. Asymptotic series of quantum field theory and their summation. Fortschr. Phys. 28 (1980) 465. https://doi.org/10.1002/prop.19800280803

3. C. Itzykson, J.-B. Zuber. Quantum Field Theory (New York: McGraw-Hill, 1980) 705 p. Google books

4. J. Zinn-Justin. Quantum Field Theory and Critical Phenomena (Oxford: Clarendon Press, 2002) 1054 p. https://doi.org/10.1093/acprof:oso/9780198509233.001.0001

5. F.T. Hioe, D. MacMillen, E.W. Montroll. Quantum theory of anharmonic oscillators. Phys. Rep. 43 (1978) 305. https://doi.org/10.1016/0370-1573(78)90097-2

6. B. Simon. Large orders and summability of eigenvalue perturbation theory: a mathematical overview. Int. J. Quant. Chem. 21 (1982) 3. https://doi.org/10.1002/qua.560210103

7. G.A. Arteca, F.M. Fernández, E.A. Castro. Large Order Perturbation Theory and Summation Methods in Quantum Mechanics (Berlin: Springer-Verlag, 1990) 644 p. https://doi.org/10.1007/978-3-642-93469-8

8. B. Simon. Fifty years of eigenvalue perturbation theory. Bull. Am. Math. Soc. 24 (1991) 303. https://doi.org/10.1090/S0273-0979-1991-16020-9

9. E.Z. Liverts, V.B. Mandelzweig, F. Tabakin. Analytic calculation of energies and wave functions of the quartic and pure quartic oscillators. J. Math. Phys. 47 (2006) 062109. https://doi.org/10.1063/1.2209769

10. H. Ezawa, M. Saito, T. Nakamura. Notes on the Pade approximation for an anharmonic oscillator. J. Phys. Soc. Japan 83 (2014) 034003. https://doi.org/10.7566/JPSJ.83.034003

11. T. Sulejmanpasic, M. Ünsal. Aspects of perturbation theory in quantum mechanics. Comput. Phys. Comm. 228 (2018) 273. https://doi.org/10.1016/j.cpc.2017.11.018

12. J. Zinn-Justin. Perturbation series at large orders in quantum mechanics and field theories: application to the problem of resummation. Phys. Rep. 70 (1981) 109. https://doi.org/10.1016/0370-1573(81)90016-8

13. J.C. Le Guillou, J. Zinn-Justin (Eds.). Large-Order Behaviour of Perturbation Theory (North Holland, Amsterdam: Elsevier Science Publishers, 1990) 594 p. Google books

14. G.A. Baker, P. Graves-Morris. Pade Approximants. 2nd ed. (Cambridge: Cambridge University Press, 1996) 764 p. https://doi.org/10.1017/CBO9780511530074

15. C.M. Bender, S.A. Orszag. Advanced Mathematical Methods for Scientists and Engineers (New York - Berlin: Springer-Verlag, 1999) 607 p. https://doi.org/10.1007/978-1-4757-3069-2

16. G.H. Hardy. Divergent Series (Oxford: Clarendon Press, 1949) 396 p. Google books

17. J.P. Ramis. Séries Divergentes et Développements Asymptotiques (Dijon: Universite de Dijon, 1993) 101 p. Google books

18. F.T. Hioe, E.W. Montroll. Quantum theory of anharmonic oscillators. J. Math. Phys. 16 (1975) 1945. https://doi.org/10.1063/1.522747

19. A.V. Turbiner. The eigenvalue spectrum in quantum mechanics and the nonlinearization procedure. Sov. Phys. Usp. 27 (1984) 668. https://doi.org/10.1070/PU1984v027n09ABEH004155

20. I.M. Suslov. Divergent perturbation series. JETP 100 (2005) 1188. https://doi.org/10.1134/1.1995802

21. B. Simon. Coupling constant analiticity for the anharmonic oscillator. Ann. Phys. 58 (1970) 76. https://doi.org/10.1016/0003-4916(70)90240-X

22. J.J. Loeffel et al. Pade approximants and the anharmonic oscillator. Phys. Lett. B 30 (1969) 656. https://doi.org/10.1016/0370-2693(69)90087-2

23. C.M. Bender, G.V. Dunne. Large-order perturbation theory for a non-Hermitian PT-symmetric Hamiltonian. J. Math. Phys. 40 (1999) 4616. https://doi.org/10.1063/1.532991

24. G.H. Hardy, J.E. Littlewood, G. Pólya. Inequalities (Cambridge: Cambridge University Press, 1934) 338 p. Google books

25. D.S. Mitrinović. Analytic Inequalities (Berlin: Springer-Verlag, 1970) 416 p. https://doi.org/10.1007/978-3-642-99970-3

26. P.S. Bullen. Handbook of Means and Their Inequalities (Berlin: Springer-Verlag, 2003) 566 p. https://doi.org/10.1007/978-94-017-0399-4

27. R.W. Hamming. Numerical Methods for Scientists and Engineers (New York: McGraw-Hill, 1962) 411 p. Google books

28. M. Abramowitz, I.A. Stegun (Eds.). Handbook of Mathematical Functions (Washington, D.C.: National Bureau of Standards, 1964) 1046 p. Google books

29. F.B. Hildebrand. Introduction to Numerical Analysis (New York: Dover Publications, 1987) 669 p. Google books

30. L.M. Milne-Thomson. The Calculus of Finite Differences (Providence: AMS, 2000) 558 p. Google books

31. F.J. Dyson. Divergence of perturbation theory in quantum electrodynamics. Phys. Rev. 85 (1952) 631. https://doi.org/10.1103/PhysRev.85.631

32. M. Cini, S. Fubini, A. Stanghellini. Fixed angle dispersion relations for nucleon-nucleon scattering. Phys. Rev. 114 (1959) 1633. https://doi.org/10.1103/PhysRev.114.1633

33. W.T.H. van Oers, J.D. Seagrave. The neutron-deuteron scattering lengths. Phys. Lett. B 24 (1967) 562. https://doi.org/10.1016/0370-2693(67)90389-9

34. V.A. Babenko, N.M. Petrov. Description of the low-energy doublet neutron-deuteron scattering in terms of parameters characterizing bound and virtual triton states. Phys. At. Nucl. 63 (2000) 1709. https://doi.org/10.1134/1.1320139

35. V.A. Babenko, N.M. Petrov. Description of scattering and of a bound state in the two-nucleon system on the basis of the Bargmann representation of the S matrix. Phys. At. Nucl. 68 (2005) 219. https://doi.org/10.1134/1.1866377

36. V.A. Babenko, N.M. Petrov. On Triplet Low-Energy Parameters of Nucleon-Nucleon Scattering. Phys. At. Nucl. 69 (2006) 1552. https://doi.org/10.1134/S1063778806090134

37. V.A. Babenko, N.M. Petrov. The P-matrix approach in a potential description of hadron-hadron interaction. Ukr. J. Phys. 32 (1987) 971. (Rus)

38. S.N. Biswas et al. Eigenvalues of λx2m anharmonic oscillators. J. Math. Phys. 14 (1973) 1190. https://doi.org/10.1063/1.1666462

39. K. Banerjee. Accurate non-perturbative solution of eigenvalue problems with application to anharmonic oscillator. Lett. Math. Phys. 1 (1976) 323. https://doi.org/10.1007/BF00398488

40. F. Vinette, J. Číźek. Upper and lower bounds of the ground state energy of an-harmonic oscillators using renormalized inner projection. J. Math. Phys. 32 (1991) 3392. https://doi.org/10.1063/1.529452

41. E.J. Weniger. A convergent renormalized strong coupling perturbation expansion for the ground state energy of the quartic, sextic, and octic anharmonic oscillator. Ann. Phys. 246 (1996) 133. https://doi.org/10.1006/aphy.1996.0023

42. C.M. Bender, T.T. Wu. Large-order behavior of perturbation theory. Phys. Rev. Lett. 27 (1971) 461. https://doi.org/10.1103/PhysRevLett.27.461

43. C.M. Bender, T.T. Wu. Anharmonic oscillator. II. A study of perturbation theory in large order. Phys. Rev. D 7 (1973) 1620. https://doi.org/10.1103/PhysRevD.7.1620

44. G. Lévai, J.M. Arias. Search for critical-point nuclei in terms of the sextic oscillator. Phys. Rev. C 81 (2010) 044304. https://doi.org/10.1103/PhysRevC.81.044304

45. A.A. Raduta, P. Buganu. Application of the sextic oscillator with a centrifugal barrier and the spheroidal equation for some X(5) candidate nuclei. J. Phys. G 40 (2013) 025108. https://doi.org/10.1088/0954-3899/40/2/025108

46. R. Budaca. Quartic oscillator potential in the γ-rigid regime of the collective geometrical model. Eur. Phys. J. A 50 (2014) 87. https://doi.org/10.1140/epja/i2014-14087-8

47. M.M. Hammad et al. Critical potentials and fluctuations phenomena with quartic, sextic, and octic anharmonic oscillator potentials. Nucl. Phys. A 1004 (2020) 122036. https://doi.org/10.1016/j.nuclphysa.2020.122036