Generalized Nonanalytic Expansions, $\mathcal{PT}$-Symmetry and Large-Order Formulas for Odd Anharmonic Oscillators

Ulrich D. Jentschura ; Andrey Surzhykov ; Jean Zinn-Justin

Symmetry, integrability and geometry: methods and applications, Tome 5 (2009)

Résumé

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.

Détail

Keywords: $\mathcal{PT}$-symmetry; asymptotics; higher-order corrections; instantons.

Bibliographie

[1] Bender C. M., Wu T. T., “Anharmonic oscillator”, Phys. Rev., 184 (1969), 1231–1260 | DOI | MR

[2] Bender C. M., Wu T. T., “Large-order behavior of perturbation theory”, Phys. Rev. Lett., 27 (1971), 461–465 | DOI

[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 | DOI

[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 | DOI | MR

[5] Le Guillou J. C., Zinn-Justin J., “Critical exponents from field theory”, Phys. Rev. B, 21 (1980), 3976–3998 | DOI | MR | Zbl

[6] Itzykson C., Zuber J. B., Quantum field theory, McGraw-Hill, New York, 1980 | MR

[7] Pachucki K., “Effective Hamiltonian approach to the bound state: Positronium hyperfine structure”, Phys. Rev. A, 56 (1997), 297–304 | DOI

[8] Erickson G. W., Yennie D. R., “Radiative level shifts. I. Formulation and lowest order Lamb shift”, Ann. Physics, 35 (1965), 271–313 | DOI | MR

[9] Erickson G. W., Yennie D. R., “Radiative level shifts. II. Higher order contributions to the Lamb shift”, Ann. Physics, 35 (1965), 447–510 | DOI | MR

[10] Karshenboim S. G., “Two-loop logarithmic corrections in the hydrogen Lamb shift”, J. Phys. B, 29 (1996), L29–L31 | DOI

[11] Pachucki K., “Logarithmic two-loop corrections to the Lamb shift in hydrogen”, Phys. Rev. A, 63 (2001), 042503, 8 pp., ages ; physics/0011044 | DOI | Zbl

[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 | DOI | MR | Zbl

[13] Oppenheimer J. R., “Three notes on the quantum theory of aperiodic fields”, Phys. Rev., 31 (1928), 66–81 | DOI | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[16] Jentschura U. D., Zinn-Justin J., “Instanton in quantum mechanics and resurgent expansions”, Phys. Lett. B, 596 (2004), 138–144 ; hep-ph/0405279 | DOI | MR

[17] Bender C. M., Boettcher S., “Real spectra in non-Hermitian Hamiltonians having $\mathcal PT$-symmetry”, Phys. Rev. Lett., 80 (1998), 5243–5246 ; physics/9712001 | DOI | MR | Zbl

[18] Bender C. M., Dunne G. V., “Large-order perturbation theory for a non-Hermitian $\mathcal PT$-symmetric Hamiltonian”, J. Math. Phys., 40 (1999), 4616–4621 ; quant-ph/9812039 | DOI | MR | Zbl

[19] Bender C. M., Boettcher S., Meisinger P. N., “$\mathcal PT$-symmetric quantum mechanics”, J. Math. Phys., 40 (1999), 2201–2229 ; quant-ph/9809072 | DOI | MR | Zbl

[20] Bender C. M., Brody D. C., Jones H. F., “Complex extension of quantum mechanics”, Phys. Rev. Lett., 89 (2002), 270401, 4 pp., ages ; Erratum Phys. Rev. Lett., 92 (2004), 119902 ; quant-ph/0208076 | DOI | MR | DOI | MR

[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 pp., ages | DOI

[22] Zinn-Justin J., Quantum field theory and critical phenomena, 3rd ed., Clarendon Press, Oxford, 1996 | MR

[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 | DOI | MR

[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 pp., ages ; arXiv:0711.1073 | DOI | MR | Zbl

[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 | DOI

[27] Pham F., “Fonctions résurgentes implicites”, C. R. Acad. Sci. Paris Sér. I Math., 309:20 (1989), 999–1004 | MR

[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., 310:4 (1990), 141–146 | MR | Zbl

[29] Candelpergher B., Nosmas J. C., Pham F., Approche de la Résurgence, Hermann, Paris, 1993 | MR | Zbl

[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., 2008:10 (2008), 019, 12 pp., ages ; arXiv:0806.1844 | DOI