Large order perturbation theory for a nonHermitian PT symmetric Hamiltonian

Dec, 1998
5 pages
Published in:
  • J.Math.Phys. 40 (1999) 4616-4621
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Abstract:
A precise calculation of the ground-state energy of the complex PT-symmetric Hamiltonian H=p2+1/4x2+iλx3H=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 λ2\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=p2+1/4x2ϵx3H=p^2+{1/4}x^2-\epsilon x^3.
Note:
  • 5 pages, ReVTeX