Resurgence in deformed genus-1 quantum-mechanical systems: the perturbative/non-perturbative relation

Perturbation theory only sees the local information about a potential, and the effects of global non-perturbative properties, such as quantum tunneling, are invisible in perturbation theory calculations. The application of resurgence theory in quantum mechanics led to the view that there are hidden relations between the perturbative and non-perturbative contributions to energy levels, a concrete example for which is the "perturbative/non-perturbative" (P/NP) relation, which allows the calculation of low-order non-perturbative instanton contributions solely from low-order perturbation theory, which can otherwise be significantly challenging to calculate. However, this relation is applicable only to genus-1 potentials with symmetries, such as double-well or Sine-Gordon potentials. We investigate deformations of genus-1 potentials that break those symmetries, and present the generalization of the P/NP relation applicable to such deformed potentials, which reduces to the previous P/NP relation at the critical values of the deformation parameters. We also show that the large-order asymptotic behavior of the perturbation series for energy levels in these deformed potentials is governed by the non-perturbative contributions, which can be calculated solely from the low-order terms in perturbation theory using the generalized P/NP relation.