Large-order perturbation theory for a non-Hermitian 𝓟𝓣-symmetric Hamiltonian
- 1 October 1999
- journal article
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 40 (10), 4616-4621
- https://doi.org/10.1063/1.532991
Abstract
A precise calculation of the ground-state energy of the complex PT-symmetric Hamiltonian $H=p^2+{1/4}x^2+i \lambda x^3$, is performed using high-order Rayleigh-Schr\"odinger perturbation theory. The energy spectrum of this Hamiltonian has recently been shown to be real using numerical methods. The Rayleigh-Schr\"odinger perturbation series is Borel summable, and Pad\'e summation provides excellent agreement with the real energy spectrum. Pad\'e analysis provides strong numerical evidence that the once-subtracted ground-state energy considered as a function of $\lambda^2$ is a Stieltjes function. The analyticity properties of this Stieltjes function lead to a dispersion relation that can be used to compute the imaginary part of the energy for the related real but unstable Hamiltonian $H=p^2+{1/4}x^2-\epsilon x^3$.Comment: 5 pages, ReVTe
Keywords
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