Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 005, 10 pages      arXiv:0901.1858      https://doi.org/10.3842/SIGMA.2009.005
Contribution to the Proceedings of the VIIth Workshop ''Quantum Physics with Non-Hermitian Operators''

Generalized Nonanalytic Expansions, PT-Symmetry and Large-Order Formulas for Odd Anharmonic Oscillators

Ulrich D. Jentschura a, Andrey Surzhykov b and Jean Zinn-Justin c
a) Department of Physics, Missouri University of Science and Technology, Rolla MO65409-0640, USA
b) Physikalisches Institut der Universität, Philosophenweg 12, 69120 Heidelberg, Germany
c) CEA, IRFU and Institut de Physique Théorique, Centre de Saclay, F-91191 Gif-Sur-Yvette, France

Received October 30, 2008, in final form January 07, 2009; Published online January 13, 2009

Abstract
The concept of a generalized nonanalytic expansion which involves nonanalytic combinations of exponentials, logarithms and powers of a coupling is introduced and its use illustrated in various areas of physics. Dispersion relations for the resonance energies of odd anharmonic oscillators are discussed, and higher-order formulas are presented for cubic and quartic potentials.

Key words: PT-symmetry; asymptotics; higher-order corrections; instantons.

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References

  1. Bender C.M., Wu T.T., Anharmonic oscillator, Phys. Rev. 184 (1969), 1231-1260.
  2. Bender C.M., Wu T.T., Large-order behavior of perturbation theory, Phys. Rev. Lett. 27 (1971), 461-465.
  3. Bender C.M., Wu T.T., Anharmonic oscillator. II. A study in perturbation theory in large order, Phys. Rev. D 7 (1973), 1620-1636.
  4. Le Guillou J.C., Zinn-Justin J., Critical exponents for the n-vector model in three dimensions from field theory, Phys. Rev. Lett. 39 (1977), 95-98.
  5. Le Guillou J.C., Zinn-Justin J., Critical exponents from field theory, Phys. Rev. B 21 (1980), 3976-3998.
  6. Itzykson C., Zuber J.B., Quantum field theory, McGraw-Hill, New York, 1980.
  7. Pachucki K., Effective Hamiltonian approach to the bound state: Positronium hyperfine structure, Phys. Rev. A 56 (1997), 297-304.
  8. Erickson G.W., Yennie D.R., Radiative level shifts. I. Formulation and lowest order Lamb shift, Ann. Physics 35 (1965), 271-313.
  9. Erickson G.W., Yennie D.R., Radiative level shifts. II. Higher order contributions to the Lamb shift, Ann. Physics 35 (1965), 447-510.
  10. Karshenboim S.G., Two-loop logarithmic corrections in the hydrogen Lamb shift, J. Phys. B 29 (1996), L29-L31.
  11. Pachucki K., Logarithmic two-loop corrections to the Lamb shift in hydrogen, Phys. Rev. A 63 (2001), 042503, 8 pages, physics/0011044.
  12. Jentschura U.D., Pachucki K., Two-loop self-energy corrections to the fine structure, J. Phys. A: Math. Gen. 35 (2002), 1927-1942, hep-ph/0111084.
  13. Oppenheimer J.R., Three notes on the quantum theory of aperiodic fields, Phys. Rev. 31 (1928), 66-81.
  14. Zinn-Justin J., Jentschura U. D., Multi-instantons and exact results. I. Conjectures, WKB expansions, and instanton interactions, Ann. Physics 313 (2004), 197-267, quant-ph/0501136.
  15. Zinn-Justin J., Jentschura U.D., Multi-instantons and exact results. II. Specific cases, higher-order effects, and numerical calculations, Ann. Physics 313 (2004), 269-325, quant-ph/0501137.
  16. Jentschura U.D., Zinn-Justin J., Instanton in quantum mechanics and resurgent expansions, Phys. Lett. B 596 (2004), 138-144, hep-ph/0405279.
  17. Bender C.M., Boettcher S., Real spectra in non-Hermitian Hamiltonians having PT-symmetry, Phys. Rev. Lett. 80 (1998), 5243-5246, physics/9712001.
  18. Bender C.M., Dunne G.V., Large-order perturbation theory for a non-Hermitian PT-symmetric Hamiltonian, J. Math. Phys. 40 (1999), 4616-4621, quant-ph/9812039.
  19. Bender C.M., Boettcher S., Meisinger P.N., PT-symmetric quantum mechanics, J. Math. Phys. 40 (1999), 2201-2229, quant-ph/9809072.
  20. Bender C.M., Brody D.C., Jones H.F., Complex extension of quantum mechanics, Phys. Rev. Lett. 89 (2002), 270401, 4 pages, Erratum, Phys. Rev. Lett. 92 (2004), 119902, quant-ph/0208076.
  21. Jentschura U.D., Surzhykov A., Zinn-Justin J., Unified treatment of even and odd anharmonic oscillators of arbitrary degree, Phys. Rev. Lett. 102 (2009), 011601, 4 pages.
  22. Zinn-Justin J., Quantum field theory and critical phenomena, 3rd ed., Clarendon Press, Oxford, 1996.
  23. Zinn-Justin J., Intégrale de chemin en mécanique quantique: Introduction, CNRS Éditions, Paris, 2003.
  24. Feinberg J., Peleg Y., Self-adjoint Wheeler-DeWitt operators, the problem of time, and the wave function of the Universe, Phys. Rev. D 52 (1995), 1988-2000, hep-th/9503073.
  25. Jentschura U.D., Surzhykov A., Lubasch M., Zinn-Justin J., Structure, time propagation and dissipative terms for resonances, J. Phys. A: Math. Theor. 41 (2008), 095302, 16 pages, arXiv:0711.1073.
  26. Benassi L., Grecchi V., Harrell E., Simon B., Bender-Wu formula and the Stark effect in hydrogen, Phys. Rev. Lett. 42 (1979), 704-707, Erratum, Phys. Rev. Lett. 42 (1979), 1430.
  27. Pham F., Fonctions résurgentes implicites, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 20, 999-1004.
  28. Delabaere E., Dillinger H., Pham F., Développements semi-classiques exacts des niveaux d'énergie d'un oscillateur à une dimension, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), no. 4, 141-146.
  29. Candelpergher B., Nosmas J.C., Pham F., Approche de la Résurgence, Hermann, Paris, 1993.
  30. Andrianov A.A., Cannata F., Giacconi P., Kamenshchik A.Y., Regoli D., Two-field cosmological models and large-scale cosmic magnetic fields, J. Cosmol. Astropart. Phys. 2008 (2008), no. 10, 019, 12 pages, arXiv:0806.1844.

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