Comment on a recent paper by Mezincescu

It has been conjectured that for ≥0 the entire spectrum of the non-Hermitian -symmetric Hamiltonian HN = p2 + x2(ix), where N = 2 + , is real. Strong evidence for this conjecture for the special case N = 3 was provided in a recent paper by Mezincescu (Mezincescu G A 2000 J. Phys. A: Math. Gen. 33 4911) in which the spectral zeta function Z3(1) for the Hamiltonian H3 = p2 + ix3 was calculated exactly. Here, the calculation of Mezincescu is generalized from the special case N = 3 to the region of all N≥2 (≥0) and the exact spectral zeta function ZN(1) for HN is obtained. Using ZN(1) it is shown that to extremely high precision (about three parts in 1018) the spectrum of HN for other values of N such as N = 4 is entirely real.