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=p2+ (1) /(4) x2+iλx3, is performed using high-order Rayleigh - Schroedinger 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 - Schroedinger perturbation series is Borel summable, and show that Padacute e 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 λ2 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=p2+ (1) /(4) and h;x2-εx3. copyright 1999 American Institute of Physics