Some properties of eigenvalues and eigenfunctions of the cubic oscillator with imaginary coupling constant

Comparison between the exact value of the spectral zeta function, $Z_{H}(1)=5^{-6/5}[3-2\cos(\pi/5)]\Gamma^2(1/5)/\Gamma(3/5)$, and the results of numeric and WKB calculations supports the conjecture by Bessis that all the eigenvalues of this PT-invariant hamiltonian are real. For one-dimensional Schr\"odinger operators with complex potentials having a monotonic imaginary part, the eigenfunctions (and the imaginary parts of their logarithmic derivatives) have no real zeros.