  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.
