Large-order perturbation theory for a non-Hermitian PT-symmetric Hamiltonian

A precise calculation of the ground-state energy of the complex PT-symmetric Hamiltonian H = p(2) + 1/4x(2) + i lambda x(3), is performed using high-order Rayleigh-Schrodinger perturbation theory. The energy spectrum of this Hamiltonian has recently been shown to be real using numerical methods. Here we present convincing numerical evidence that the Rayleigh-Schrodinger perturbation series is Borel summable, and show that Pade summation provides excellent agreement with the real energy spectrum. Pade 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). (C) 1999 American Institute of Physics. [S0022-2488(99)01810-1].