On the reality of the eigenvalues for a class of PT-symmetric oscillators

We study the eigenvalue problem -u"(z)-[(iz)^m+P(iz)]u(z)=\lambda u(z) with the boundary conditions that u(z) decays to zero as z tends to infinity along the rays \arg z=-\frac{\pi}{2}\pm \frac{2\pi}{m+2}, where P(z)=a_1 z^{m-1}+a_2 z^{m-2}+...+a_{m-1} z is a real polynomial and m\geq 2. We prove that if for some 1\leq j\leq\frac{m}{2}, we have (j-k)a_k\geq 0 for all 1\leq k\leq m-1, then the eigenvalues are all positive real. We then sharpen this to a slightly larger class of polynomial potentials. In particular, this implies that the eigenvalues are all positive real for the potentials \alpha iz^3+\beta z^2+\gamma iz when \alpha,\beta and \gamma are all real with \alpha\not=0 and \alpha \gamma \geq 0, and with the boundary conditions that u(z) decays to zero as z tends to infinity along the positive and negative real axes. This verifies a conjecture of Bessis and Zinn-Justin.