Non-Perturbative Quantum Mechanics from Non-Perturbative Strings

TL;DR

This paper introduces a novel approach using trans-series solutions to refined holomorphic anomaly equations for calculating non-perturbative effects in quantum mechanics, extending existing methods and providing new insights into spectral problems.

Contribution

It develops a new non-perturbative calculation method based on topological string theory, applicable to Schrödinger problems and mirror curve quantizations, extending prior approaches like the exact WKB method.

Findings

Successfully reproduces known non-perturbative results

Provides new trans-series structures for mirror curve spectra

Verifies predictions against large-order behavior of perturbative sectors

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 Schr\"odinger 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.