Comment on a recent paper by Mezincescu (Q2735717)

[This comment concerns \textit{G. A. Mezincescu}, J. Phys. A, Math. Gen. 33, 4911-4966 (2000; Zbl 0973.34070).]NEWLINENEWLINENEWLINEIt has been conjectured that for \(\varepsilon\geq 0\) the entire spectrum of the non-Hermitian \({\mathcal P}{\mathcal T}\)-symmetric Hamiltonian \(H_N=p^2+x^2 (ix)^\varepsilon \), where \(N=2+\varepsilon\), is real. Strong evidence for this conjecture for the special case \(N=3\) was provided by Mezincescu (loc. cit.) commented here in which the spectral zeta function \(Z_3(1)\) for the Hamiltonian \(H_3=p^2 +ix^3\) was calculated exactly. Here, the calculation of Mezincescu is generalized from the special case \(N=3\) to the region of all \(N\geq 2(\varepsilon \geq 0)\) 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 precision (about three parts in \(10^{18})\) the spectrum of \(H_N\) for other values of \(N\) such as \(N=4\) is entirely real.