Numerical estimates of the spectrum for anharmonic PT symmetric potentials

In this work, we wish to revisit the energy spectrum for the anharmonic potentials H=ρ2/(2μ)+sxN, where N = 2,3,4,6,8 and s = -1,i,+1. By expanding the Hamiltonian matrices in terms of simple harmonic oscillator (SHO) states, we have numerically evaluated the energy eigenvalue spectrum of matrix truncations of up to 1000 × 1000. By extrapolating the minimum and the maximum of the energy spectrum as a function of matrix size, we are then able to determine the energy spectral range in the thermodynamic limit. Our results for the energy spectra are in disagreement with the work of Bender and Boettcher (1998 Phys. Rev. Lett. 80 5243) who studied a class of Hamiltonians with potential energies of the form V (x) = + x2(ix)r. They maintain that when r ⩾ 0, the spectrum is discrete, entirely real and positive extending to infinity. We find that the energy spectrum for potentials with r = 0,1,2,6,8 is not consistent with those results, whereas for r = 4 the spectrum is positive and discrete. Questions are raised about methods used by Bender to determine the energy spectrum as a function of r. (paper)