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 [3-2cos (π/5)]Γ 2 ((1/5))/Γ((3/5)), and the results of numeric and WKB calculations supports the conjecture by Daniel Bessis (1995 private communication) that all the eigenvalues of this PT -invariant Hamiltonian are real. For one-dimensional Schrödinger operators with complex potentials having a monotonic imaginary part, the eigenfunctions (and the imaginary parts of their logarithmic derivatives) have no real zeros.