A Note on Instantons in a 1D Same-Level Asymmetric Double Well
Klaus Bering
TL;DR
The paper extends Coleman's 1D instanton framework to same-level asymmetric double wells, where neighboring minima can have different Hessians and frequencies, and derives all-order multi-instanton corrections in the Euclidean path integral. A key methodological advance is the introduction of a step-function reference potential for the Gelfand-Yaglom determinant, enabling a relative determinant calculation that accounts for frequency mismatches. The results yield explicit two-level-system descriptions for both odd and even instanton sectors, with energy levels $E_±$ and tunneling-induced splittings $\Delta E$, expressed in terms of the action $\bar{S}$ and fluctuation factor $K$, including a determinant ratio $K_0$. The paper also provides concrete symmetric and same-level triple-well examples, showing that the all-order instanton corrections deform the two lowest energies while preserving a two-level (or three-level in the triple-well) interpretation, and highlighting the practical relevance for quantum-mechanical tunneling in structured potentials.
Abstract
We prove formulas for the multi-instanton corrections to the overlap and energies of a 1D same-level asymmetric double well using the Euclidean path integral. Both the odd and even instanton sectors are summed to all orders. The double well is same-level asymmetric in the sense that the potentials at neighboring wells have the same bottom level but can have different Hessians/curvatures/frequencies, which modify Coleman's original formulas. This for instance implies that the reference model used to calculate the functional determinant of quantum fluctuations must now interpolate between simple harmonic oscillators of different frequencies. Examples of symmetric double and triple wells are worked out.