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

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.

11 pages; 12 figures

References in corpus (4)

• Making Sense of Non-Hermitian Hamiltonians
• Emergence of Tricritical Point and Liquid-Gas Phase in the Massless 2+1 Dimensional Gross-Neveu Model
• Asymptotically Improved Convergence of Optimized Perturbation Theory in the Bose-Einstein Condensation Problem
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