Exact instanton transseries for quantum mechanics

TL;DR

This paper derives exact instanton transseries for quantum mechanical oscillators, unifying nonperturbative corrections in a closed form and clarifying their resurgent structure using alien calculus.

Contribution

It introduces a minimal one-parameter transseries for nonperturbative energy corrections, applicable to various quantum oscillators, and elucidates the Stokes phenomenon and transseries structure.

Findings

Exact all-order instanton corrections for quantum oscillators

Unified transseries description capturing Stokes phenomena

Factorization of complex transseries in cosine potential

Abstract

We calculate the instanton corrections to energy spectra of one-dimensional quantum mechanical oscillators to all orders and unify them in a closed form transseries description. Using alien calculus, we clarify the resurgent structure of these transseries and demonstrate two approaches in which the Stokes constants can be derived. As a result, we formulate a minimal one-parameter transseries for the natural nonperturbative extension to the perturbative energy, which captures the Stokes phenomenon in a single stroke. We derive these results in three models: quantum oscillators with cubic, symmetric double well and cosine potentials. In the latter two examples, we find that the resulting full transseries for the energy has a more convoluted structure that we can factorise in terms of a minimal and a median transseries. For the cosine potential we briefly discuss this more complicated transseries structure in conjunction with topology and the concept of the resurgence triangle.