Comment on a recent paper by Mezincescu

It has been conjectured that for epsilon greater than or equal to 0 the ent ire spectrum of the non-Hermitian PT-symmetric Hamiltonian H-N = p(2) + x(2 )(ix)(epsilon), where N = 2 + epsilon, is real. Strong evidence for this co njecture for the special case N = 3 was provided in a recent paper by Mezin cescu (Mezincescu G A 2000 J. Phys. A: Math. Gen. 33 4911) in which the spe ctral zeta function Z(3)(1) for the Hamiltonian H-3 = p(2) + ix(3) was calc ulated exactly. Here, the calculation of Mezincescu is generalized from the special case N = 3 to the region of all N greater than or equal to 2 (epsi lon greater than or equal to 0) and the exact spectral zeta function Z(N)(1 ) for H-N is obtained. Using Z(N)(1) it is shown that to extremely high pre cision (about three parts in 10(18)) the spectrum of H-N for Other values o f N such as N = 4 is entirely real.