A long time ago, it has been conjectured that a Hamiltonian with a potential of the form x 2+ivx 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 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.
J. Zinn-Justin and U. D. Jentschura, "Order-Dependent Mappings: Strong-Coupling Behavior from Weak-Coupling Expansions in Non-Hermitian Theories," Journal of Mathematical Physics, vol. 51, no. 7, pp. 072106-1-072106-19, American Institute of Physics (AIP), Jul 2010.
The definitive version is available at https://doi.org/10.1063/1.3451104
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01 Jul 2010