Multi-instantons and exact results III: Unification of even and odd anharmonic oscillators

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

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This is the third article in a series of three papers on the resonance energy levels of anharmonic oscillators. Whereas the first two papers mainly dealt with double-well potentials and modifications thereof [see J. Zinn-Justin, U.D. Jentschura, Ann. Phys. (N.Y.) 313 (2004) 197 and 269], we here focus on simple even and odd anharmonic oscillators for arbitrary magnitude and complex phase of the coupling parameter. A unification is achieved by the use of PT-symmetry inspired dispersion relations and generalized quantization conditions that include instanton configurations. Higher-order formulas are provided for the oscillators of degrees 3 to 8, which lead to subleading corrections to the leading factorial growth of the perturbative coefficients describing the resonance energies. Numerical results are provided, and higher-order terms are found to be numerically significant. The resonances are described by generalized expansions involving intertwined nonanalytic exponentials, logarithmic terms and power series. Finally, we summarize spectral properties and dispersion relations of anharmonic oscillators, and their interconnections. The purpose is to look at one of the classic problems of quantum theory from a new perspective, through which we gain systematic access to the phenomenologically significant higher-order terms.

Original languageEnglish
Pages (from-to)1135-1172
Number of pages38
JournalAnnals of Physics
Issue number5
Publication statusPublished - May 1 2010



  • Asymptotic problems and properties
  • General properties of perturbation theory
  • Summation of perturbation theory

ASJC Scopus subject areas

  • Physics and Astronomy(all)

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