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Non-perturbative Quantum Mechanics from Non-perturbative Strings

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Abstract

This work develops a new method to calculate non-perturbative corrections in one-dimensional Quantum Mechanics, based on trans-series solutions to the refined holomorphic anomaly equations of topological string theory. The method can be applied to traditional spectral problems governed by the Schrödinger equation, where it both reproduces and extends the results of well-established approaches, such as the exact WKB method. It can be also applied to spectral problems based on the quantization of mirror curves, where it leads to new results on the trans-series structure of the spectrum. Various examples are discussed, including the modified Mathieu equation, the double-well potential and the quantum mirror curves of local \({\mathbb {P}}^2\) and local \({\mathbb {F}}_0\). In all these examples, it is verified in detail that the trans-series obtained with this new method correctly predict the large-order behavior of the corresponding perturbative sectors.

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Authors and Affiliations

  1. Département de Physique Théorique and Section de Mathématiques, Université de Genève, 1211, Geneve, Switzerland

    Santiago Codesido & Marcos Mariño

  2. Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA, 93106, USA

    Ricardo Schiappa

  3. CAMGSD, Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001, Lisbon, Portugal

    Ricardo Schiappa

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  1. Santiago Codesido
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  2. Marcos Mariño
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Correspondence to Marcos Mariño.

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Codesido, S., Mariño, M. & Schiappa, R. Non-perturbative Quantum Mechanics from Non-perturbative Strings. Ann. Henri Poincaré 20, 543–603 (2019). https://doi.org/10.1007/s00023-018-0751-x

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