Large-order Perturbation Theory for a Non-Hermitian PT-symmetric Hamiltonian

Details Large-order Perturbation Theory for a Non-Hermitian PT-symmetric Hamiltonian Large-order Perturbation Theory for a Non-Hermitian PT-symmetric Hamiltonian

1998-12-16

arxiv.org/abs/quant-ph/9812039v1

J.Math.Phys. 40 (1999) 4616-4621

Physics

Quantum Physics

5 pages, ReVTeX

Scientific paper

10.1063/1.532991

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 $H=p^2+{1/4}x^2-\epsilon x^3$.

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