Multi-Instantons and Exact Results III: Unified Description of the Resonances of Even and Odd Anharmonic Oscillators

TL;DR

This paper presents a unified approach to understanding resonance energies in even and odd anharmonic oscillators using PT-symmetry, instanton configurations, and generalized quantization, providing higher-order corrections and numerical analysis.

Contribution

It introduces a unified framework for resonance energies of anharmonic oscillators incorporating instanton effects and higher-order terms, extending previous double-well analyses.

Findings

Higher-order formulas for oscillators of degrees 3 to 8.

Numerical significance of higher-order corrections.

Complex resonance structures involving non-analytic expansions.

Abstract

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 and U. D. Jentschura, Ann. Phys. (N.Y.) 313 (2004), pp. 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 non-analytic 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.