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Resurgence Theory and Holomorphic Quantum Mechanics

M. W. AlMasri

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.

Resurgence Theory and Holomorphic Quantum Mechanics

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 () up to sixth order in the coupling ; the resulting exact rational coefficients reproduce the classic Bender--Wu results, confirming the consistency and power of the holomorphic resurgence approach.

Paper Structure

This paper contains 6 sections, 3 theorems, 100 equations, 1 table.

Table of Contents

  1. Introduction
  2. Formalism
  3. Holomorphic Quantum Mechanics
  4. Resurgence Program
  5. Results
  6. Conclusion

Key Result

Lemma 2

The Segal--Bargmann space $\mathcal{H}_{\mathrm{SB}}$ is unitarily isomorphic to the Schrödinger Hilbert space $L^{2}(\mathbb{R})$. The unitary isomorphism is given by the Segal--Bargmann transform which satisfies and extends uniquely to a unitary operator $T : L^{2}(\mathbb{R}) \to \mathcal{H}_{\mathrm{SB}}$.

Theorems & Definitions (5)

  • Definition 1: Segal--Bargmann space
  • Lemma 2: Unitary equivalence via the Segal--Bargmann transform
  • Lemma 3: Reproducing kernel of the Segal--Bargmann space
  • Definition 4: Toeplitz operator
  • Theorem 5: Ladder operators as Toeplitz operators Hall