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

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 foc...

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Bibliographic Details
Published in:Annals of physics 2010-05, Vol.325 (5), p.1135-1172
Main Authors: Jentschura, Ulrich D., Surzhykov, Andrey, Zinn-Justin, Jean
Format: Article
Language:English
Subjects:
ANHARMONIC OSCILLATORS
Asymptotic problems and properties
ASYMPTOTIC SOLUTIONS
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
DISPERSION RELATIONS
ENERGY LEVELS
General properties of perturbation theory
INSTANTONS
OSCILLATORS
PERTURBATION THEORY
POWER SERIES
QUANTIZATION
Quantum physics
Quantum theory
RESONANCE
Summation of perturbation theory
SYMMETRY
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Summary: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.
ISSN:0003-4916
1096-035X
DOI:10.1016/j.aop.2010.01.002