High order perturbation theory for difference equations and Borel summability of quantum mirror curves

Abstract

We adapt the Bender-Wu algorithm 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.

39 pages, 4 figures, and 4 tables. Bundled with the source files of this document are the Mathematica notebooks for the package BenderWu, including the new function BWDifference

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