High order perturbation theory for difference equations and Borel summability of quantum mirror curves. We adapt the Bender-Wu algorithm C. M. Bender and T. T. Wu, “Anharmonic oscillator. 2: A study of perturbation theory in large order” in [Phys. Rev. D 7, 1620–1636 (1973; doi:1103/PhysRevD.7.1620)] to solve perturbatively but very efficiently the eigenvalue problem of “relativistic” quantum mechanical problems whose Hamiltonians are difference operators of the exponential-polynomial type. We implement the algorithm in the function BWDifference in the updated Mathematica package BenderWu. With the help of BWDifference, we survey quantum mirror curves of toric fano Calabi-Yau threefolds, and find strong evidence that not only are the perturbative eigenenergies of the associated 1d quantum mechanical problems Borel summable, but also that the Borel sums are exact.

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