Resurgence of deformed genus-1 curves: A novel perturbative/nonperturbative relation

TL;DR

This paper introduces a new perturbative/nonperturbative relation applicable to a wide range of genus-1 potentials, enhancing the ability to compute nonperturbative effects in quantum systems with complex instantons.

Contribution

It extends the P/NP relation to include polynomial and elliptic potentials with real and complex instantons, broadening the scope of quantum systems where perturbation theory is effective.

Findings

Predicted corrections to large-order behavior of perturbative series for Jacobi SD elliptic potential.

Validated the P/NP relation using early instanton fluctuation terms.

Expanded understanding of nonperturbative effects in genus-1 quantum potentials.

Abstract

We present a new perturbative/nonperturbative (P/NP) relation that applies to a broader class of genus-1 potentials, including those that possess real and complex instantons parametrized by a deformation parameter, such as polynomial and elliptic potentials. Our findings significantly extend the scope of quantum mechanical systems for which perturbation theory suffices to calculate the contributions of nonperturbative effects to energy levels, with or without the need for a boundary condition, depending on the potential. We further provide evidence for our results by predicting the corrections to the large-order behavior of the perturbative expansion for the Jacobi SD elliptic potential using the early terms of the instanton fluctuations that satisfy the P/NP relation.