Resurgence Theory and Holomorphic Quantum Mechanics

TL;DR

This paper explores the resurgence program in holomorphic quantum mechanics, focusing on the quartic anharmonic oscillator, and demonstrates how non-perturbative effects can be explicitly connected to perturbative series through the instanton operator.

Contribution

It introduces a holomorphic resurgence framework for quantum mechanics, explicitly relating perturbative and non-perturbative sectors using the Segal--Bargmann space and instanton operators.

Findings

Perturbative energy series is Gevrey-1 and Borel summable after Stokes line continuation.

The instanton operator provides an explicit link between perturbative coefficients and non-perturbative sectors.

Computed first seven energy levels, matching classic Bender--Wu results.

Abstract

In this work, we study the resurgence program in holomorphic quantum mechanics. As a specific problem, we investigate the resurgence in the quartic anharmonic oscillator within holomorphic quantum mechanics, using the Bargmann representation of bosonic operators. In this framework, the perturbative energy series is shown to be Gevrey-1 and Borel summable only after continuation across the Stokes line. The instanton operator, realized as a coherent-state displacement in the Segal--Bargmann space, provides an explicit operatorial bridge between perturbative coefficients and non-perturbative sectors. Alien derivative relations generate the full resurgence triangle characteristic of the Bender--Wu model, and the resummed energy is expressed as a trans-series via a ratio of expectation values involving this instanton operator. As a concrete demonstration, we compute the first seven energy levels ($n=0,\dots,6$) up to sixth order in the coupling $g$; the resulting exact rational coefficients reproduce the classic Bender--Wu results, confirming the consistency and power of the holomorphic resurgence approach.