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+14x2+iλx3, 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. Here we present convincing numerical evidence that the Rayleigh–Schrödinger perturbation series is Borel summable, and show that 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 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+14 x2−εx3.