Multi-Instantons and Exact-WKB Stokes Jumps in Multi-Well Quantum Mechanics
Abstract
We study the relation between the exact-WKB Delabaere--Dillinger--Pham Stokes jumps and the multi-instanton expansion of a one-dimensional potential with N equal-depth minima. For a single barrier this relation is the Bogomolny--Zinn-Justin cancellation between the instanton--anti-instanton ambiguity and the Borel ambiguity of perturbation theory, and we extend it to every consecutive barrier interval and to the integration contours themselves. In the dilute large-T regime at fixed nonperturbative grade, we derive the quasi-zero mode integral over the instanton separations from the Euclidean path integral and reduce it to Gamma and Hankel periods. With the local Weber level function of each well and one normalization per well fixed by the local Weber connection matrix, these periods reproduce the two lateral connected exact-WKB coefficients. The two lateral contours of each instanton--anti-instanton separation differ by exactly one Hankel cycle, and this linear Picard--Lefschetz relation reproduces the Stokes jump of the connected coefficient. A numerical solution of the Schrödinger equation for a symmetric triple well confirms the resulting two-barrier level splitting. At higher fixed grade, the map from ordered event sequences to normalized scalar coefficient products is not one-to-one. At leading dilute order the scalar weight of a closed event sequence depends only on the numbers of barrier crossings, ℐℐ̅ pairs, and ℐℐ/ℐ̅ℐ̅ pairs. We construct two inequivalent grade-six triple-well event sequences with the same product of local Weber amplitudes, which embed in every potential with N≥3 wells. Our results hold within the regular dilute, one-loop, local Weber approximation.
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BibTeX
@misc{behtash2026multiinstantons,
title = {{Multi-Instantons and Exact-WKB Stokes Jumps in Multi-Well Quantum Mechanics}},
author = {Alireza Behtash and Syo Kamata},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20006},
eprint = {2610.20006},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20006},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Multi-Instantons and Exact-WKB Stokes Jumps in Multi-Well Quantum Mechanics
AU - Alireza Behtash
AU - Syo Kamata
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20006
UR - https://arxiv.org/abs/2610.20006
AB - We study the relation between the exact-WKB Delabaere--Dillinger--Pham Stokes jumps and the multi-instanton expansion of a one-dimensional potential with N equal-depth minima. For a single barrier this relation is the Bogomolny--Zinn-Justin cancellation between the instanton--anti-instanton ambiguity and the Borel ambiguity of perturbation theory, and we extend it to every consecutive barrier interval and to the integration contours themselves. In the dilute large-T regime at fixed nonperturbative grade, we derive the quasi-zero mode integral over the instanton separations from the Euclidean path integral and reduce it to Gamma and Hankel periods. With the local Weber level function of each well and one normalization per well fixed by the local Weber connection matrix, these periods reproduce the two lateral connected exact-WKB coefficients. The two lateral contours of each instanton--anti-instanton separation differ by exactly one Hankel cycle, and this linear Picard--Lefschetz relation reproduces the Stokes jump of the connected coefficient. A numerical solution of the Schrödinger equation for a symmetric triple well confirms the resulting two-barrier level splitting. At higher fixed grade, the map from ordered event sequences to normalized scalar coefficient products is not one-to-one. At leading dilute order the scalar weight of a closed event sequence depends only on the numbers of barrier crossings, ℐℐ̅ pairs, and ℐℐ/ℐ̅ℐ̅ pairs. We construct two inequivalent grade-six triple-well event sequences with the same product of local Weber amplitudes, which embed in every potential with N≥3 wells. Our results hold within the regular dilute, one-loop, local Weber approximation.
ER -