On the reality of the eigenvalues for a class of \(\mathcal{PT}\)-symmetric oscillators (Q1849417)

Here, the author studies the eigenvalue problem for the equation \[ -u''(z)-[(iz)^{m}+P(iz)]u(z) = \lambda u(z) \] 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 \(m\geq 2\). It is proved that, if for some \(1\leq j\leq\frac{m}{2}\) one has \((j-k)a_{k}\geq 0\) for all \(1\leq k\leq m-1\), then the eigenvalues are all positive real.