Unify the effect of anharmonicity in double-wells and anharmonic oscillators

TL;DR

This paper investigates the effects of anharmonicity in quantum oscillators and double-well potentials, proposing a unified qualitative formula for energy gaps across all coupling regimes, supported by numerical bootstrap analysis.

Contribution

It introduces a unified formula for energy gaps in anharmonic systems, linking oscillators and double-wells, and explains this connection through quantum phase transitions.

Findings

Proposes a qualitative formula for energy gaps valid for all coupling strengths.

Shows the formula's functional similarity to double-well ground state splitting.

Numerically justifies the connection up to octic anharmonicities.

Abstract

We study the effect of anharmonicity in quantum anharmonic oscillators, by computing the energy gap between the ground and the 1st excited state using the numerical bootstrap method. Based on perturbative formulae of limiting coupling regimes, we propose a qualitative formula of the energy gap across all coupling values. Except detailed numerical parameter values, the proposed formula has the same functional form as the formula of ground state level splitting of double-well potentials, proposed recently in arXiv:2308.11516. This unifies the effects of anharmonicity in both the anharmonic oscillators and the double-well potentials, although the underlying physical process of them are completely different. We give an explanation of this connection of their anharmonicity from the viewpoint of quantum phase transitions. This connection is justified up to the octic anharmonicities by the numerical bootstrap.