  We study the eigenvalue problem
$-u^{\prime\prime}(z)-[(iz)^m+P_{m-1}(iz)]u(z)=\lambda u(z)$ 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 $P_{m-1}(z)=a_1 z^{m-1}+a_2
z^{m-2}+...+a_{m-1} z$ is a polynomial and integers $m\geq 3$. We provide an
asymptotic expansion of the eigenvalues $\lambda_n$ as $n\to+\infty$, and prove
that for each {\it real} polynomial $P_{m-1}$, the eigenvalues are all real and
positive, with only finitely many exceptions.
