Order-dependent mappings: Strong-coupling behavior from weak-coupling expansions in non-Hermitian theories

Journal Article · Thu Jul 15 00:00:00 EDT 2010 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.3451104· OSTI ID:21362171

Zinn-Justin, Jean ^[1]; Jentschura, Ulrich D ^[2]

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1. CEA, IRFU, Centre de Saclay, 91191 Gif-sur-Yvette Cedex (France)
2. Missouri University of Science and Technology, Rolla, Missouri 65409-0640 (United States)

A long time ago, it has been conjectured that a Hamiltonian with a potential of the form x{sup 2}+ivx{sup 3}, v real, has a real spectrum. This conjecture has been generalized to a class of the 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{sup 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 ODM 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.

• OSTI ID: 21362171
• Journal Information: Journal of Mathematical Physics, Vol. 51, Issue 7; Other Information: DOI: 10.1063/1.3451104; (c) 2010 American Institute of Physics; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICAL METHODS AND COMPUTING
ACCURACY
CONVERGENCE
DISTURBANCES
FEYNMAN DIAGRAM
HAMILTONIANS
HERMITIAN MATRIX
MAPPING
PERTURBATION THEORY
POTENTIALS
RENORMALIZATION
SINGULARITY
SPECTRA
STRONG-COUPLING MODEL
SYMMETRY
TRANSFORMATIONS
DIAGRAMS
INFORMATION
MATHEMATICAL MODELS
MATHEMATICAL OPERATORS
MATRICES
PARTICLE MODELS
QUANTUM OPERATORS