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

TL;DR

This paper develops an efficient perturbative solution method for quantum difference equations and provides evidence that the eigenenergies of quantum mirror curves are Borel summable and exact.

Contribution

It adapts the Bender-Wu algorithm for difference operators and applies it to quantum mirror curves of toric Fano Calabi-Yau threefolds, revealing Borel summability of eigenenergies.

Findings

Perturbative eigenenergies are Borel summable.

Borel sums of eigenenergies are exact.

The method is implemented in the BenderWu Mathematica package.

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.