Imaginary Cubic Perturbation: Numerical and Analytic Study

TL;DR

This paper explores the analytic continuation and convergence properties of the ground state resonance energy in the cubic potential, using resummed expansions and order-dependent mappings to connect weak and strong coupling regimes.

Contribution

It introduces a novel approach to analytically continue and analyze the resonance energy of the cubic potential across different coupling regimes using resummation and ODM techniques.

Findings

Successful analytic continuation to the second Riemann sheet.

Identification of resonance-antiresonance merging at a critical point.

Effective interpolation between weak and strong coupling regimes.

Abstract

The analytic properties of the ground state resonance energy E(g) of the cubic potential are investigated as a function of the complex coupling parameter g. We explicitly show that it is possible to analytically continue E(g) by means of a resummed strong coupling expansion, to the second sheet of the Riemann surface, and we observe a merging of resonance and antiresonance eigenvalues at a critical point along the line arg(g) = 5 pi/4. In addition, we investigate the convergence of the resummed weak-coupling expansion in the strong coupling regime, by means of various modifications of order-dependent mappings (ODM), that take special properties of the cubic potential into account. The various ODM are adapted to different regimes of the coupling constant. We also determine a large number of terms of the strong coupling expansion by resumming the weak-coupling expansion using the ODM, demonstrating the interpolation between the two regimes made possible by this summation method.