Order-dependent mappings: strong coupling behaviour from weak coupling expansions in non-Hermitian theories

TL;DR

This paper demonstrates that order-dependent mappings can effectively sum divergent perturbation series in non-Hermitian PT-symmetric Hamiltonians, enabling insights into strong-coupling behavior and singularity locations.

Contribution

It introduces an ODM summation method that accurately extends weak-coupling expansions to strong-coupling regimes in non-Hermitian theories.

Findings

Efficient summation of divergent series across the complex plane.

Identification of level crossing singularities.

Accurate approximation of strong-coupling limits.

Abstract

A long time ago, it has been conjectured that a Hamiltonian with a potential of the form x^2+i v x^3, v real, has a real spectrum. This conjecture has been generalized to a class of so-called PT symmetric Hamiltonians and some proofs have been given. Here, we show by numerical investigation that the divergent perturbation series can be summed efficiently by an order-dependent mapping (ODM) in the whole complex plane of the coupling parameter v^2, and that some information about the location of level crossing singularities can be obtained in this way. Furthermore, we discuss to which accuracy the strong-coupling limit can be obtained from the initially weak-coupling perturbative expansion, by the ODM summation method. The basic idea of the ODM summation method is the notion of order-dependent "local" disk of convergence and analytic continuation by an order-dependent mapping of the domain of analyticity augmented by the local disk of convergence onto a circle. In the limit of vanishing local radius of convergence, which is the limit of high transformation order, convergence is demonstrated both by numerical evidence as well as by analytic estimates.