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(pi/5)]Gamma(2)(1/5)/ Gamma(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 Schrodinger operators with complex potentials having a monotonic imaginary part, the eigenfunctions (and the imaginary parts of their logarithmic derivatives) have no real zeros.