Some properties of eigenvalues and eigenfunctions of the cubic oscillator with imaginary coupling constant
Abstract
Comparison between the exact value of the spectral zeta function, ZH(1)=5-6/5[3-2(π/5)]2(1/5)/(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.
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