Abstract
Certain quantum mechanical systems with a discrete spectrum, whose observables are given by a transseries in ħ, were shown to admit ħ0-deformations with Borel resummable expansions which reproduce the original model at ħ0 = ħ. Such expansions were dubbed Exact Perturbation Theory (EPT). We investigate how the above results can be obtained within the framework of the exact WKB method by studying the spectrum of polynomial quantum mechanical systems. Within exact WKB, energy eigenvalues are determined by exact quantization conditions defined in terms of Voros symbols \( {a}_{\gamma_i} \), γi being their associated cycles, and generally give rise to transseries in ħ. After reviewing how the Borel summability of energy eigenvalues in the quartic anharmonic potential emerges in exact WKB, we extend it to higher order anharmonic potentials with quantum corrections. We then show that any polynomial potential can be ħ0-deformed to a model where the exact quantization condition reads simply aγ = −1 and leads to the EPT Borel resummable series for all energy eigenvalues.
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Acknowledgments
We thank Alba Grassi, Alexander van Spaendonck and André Voros for discussions. We thank the organizers of the 2022 workshop “Physical resurgence: On quantum, gauge, and stringy” held at the Isaac Newton Institute in Cambridge and the 2023 workshop “Quantization and Resurgence” held at the SwissMAP Research Station in Les Diablerets for the hospitality, and the participants of both workshops for interesting discussions. MS thanks Veronica Fantini and Maxim Kontsevich for discussions, Yoshitsugu Takei for a useful e-mail correspondence, Syo Kamata for clarifications on some of the results of [27]. BB thanks SISSA for the hospitality while carrying out part of this work. MS thanks the Institut des Hautes Études Scientifiques (IHES) for the hospitality during the completion of this project. TR is partially supported by the ERC-COG grant NP-QFT No. 864583 “Non-perturbative dynamics of quantum fields: from new deconfined phases of matter to quantum black holes”, by the MIUR-SIR grant RBSI1471GJ, by the MIUR-PRIN contract 2015 MP2CX4. BB is partially supported by INFN Iniziativa Specifica TPPC. TR and MS are partially supported by INFN Iniziativa Specifica ST&FI.
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Bucciotti, B., Reis, T. & Serone, M. An anharmonic alliance: exact WKB meets EPT. J. High Energ. Phys. 2023, 124 (2023). https://doi.org/10.1007/JHEP11(2023)124
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DOI: https://doi.org/10.1007/JHEP11(2023)124