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.
Highlights
WKB [12,13,14,15,16], exact WKB [17,18,19,20]
We mention a fresh perspective on the problem of Borel summation [21, 22] in which it was shown that in quantum mechanics perturbation theory can be recast in a form which completely captures nonperturbative physics
Unsal and Argyres [24, 25] conjectured that renormalon singularities have a semi-classical explanation if the problem is approached from the regime of weakly coupled theory via the idea of adiabatic continuity [26,27,28]. In such regimes it was shown that renormalon singularities disappear [29], and resurgence is likely operative. This is difficult to test as no access to high orders of perturbation theory is typically available in QFTs
Summary
We first describe the Bender-Wu algorithm adapted to solve the eigenvalue problems of Hamiltonian difference operators, and explain how to use the function BWDifference in the BenderWu package which implements the adapted Bender-Wu algorithm
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
