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

Carl M. Bender·Gerald V. Dunne
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Abstract

A precise calculation of the ground-state energy of the complex PT-symmetric Hamiltonian $H=p^2+{1/4}x^2+i λx^3$, is performed using high-order Rayleigh-Schrödinger perturbation theory. The energy spectrum of this Hamiltonian has recently been shown to be real using numerical methods. The Rayleigh-Schrödinger perturbation series is Borel summable, and Padé summation provides excellent agreement with the real energy spectrum. Padé analysis provides strong numerical evidence that the once-subtracted ground-state energy considered as a function of $λ^2$ 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 $H=p^2+{1/4}x^2-εx^3$.

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Carl M. Bender, Gerald V. Dunne. 1998-12-16. Large-order Perturbation Theory for a Non-Hermitian PT-symmetric Hamiltonian. https://doi.org/10.1063/1.532991

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