A Note on Instantons in a 1D Same-Level Asymmetric Double Well

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.

16 pages, 2 figures, LaTeX. v2,v3: Added explanation. v3: Published version

arXiv:2505.14557 [quant-ph]

(or arXiv:2505.14557v3 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2505.14557 Focus to learn more arXiv-issued DOI via DataCite

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