Unified Treatment of Even and Odd Anharmonic Oscillators of Arbitrary Degree

TL;DR

This paper develops a unified method to analyze both even and odd anharmonic oscillators of any degree, incorporating higher-order corrections and instanton effects, advancing understanding of their spectral properties.

Contribution

It introduces a unified approach using dispersion relations and generalized quantization conditions to treat arbitrary anharmonic oscillators, including new explicit results.

Findings

Behavior of large-order perturbation theory for odd oscillators

Subleading corrections to decay widths of excited states

Explicit results for arbitrary levels of odd anharmonic oscillators

Abstract

We present a unified treatment, including higher-order corrections, of anharmonic oscillators of arbitrary even and odd degree. Our approach is based on a dispersion relation which takes advantage of the PT-symmetry of odd potentials for imaginary coupling parameter, and of generalized quantization conditions which take into account instanton contributions. We find a number of explicit new results, including the general behaviour of large-order perturbation theory for arbitrary levels of odd anharmonic oscillators, and subleading corrections to the decay width of excited states for odd potentials, which are numerically significant.