  For integers $m\geq 3$ and $1\leq\ell\leq m-1$, we study the eigenvalue
problems $-u^{\prime\prime}(z)+[(-1)^{\ell}(iz)^m-P(iz)]u(z)=\lambda u(z)$ with
the boundary conditions that $u(z)$ decays to zero as $z$ tends to infinity
along the rays $\arg z=-\frac{\pi}{2}\pm \frac{(\ell+1)\pi}{m+2}$ in the
complex plane, where $P$ is a polynomial of degree at most $m-1$. We provide
asymptotic expansions of the eigenvalues $\lambda_{n}$. Then we show that if
the eigenvalue problem is $\mathcal{PT}$-symmetric, then the eigenvalues are
all real and positive with at most finitely many exceptions. Moreover, we show
that when $\gcd(m,\ell)=1$, the eigenvalue problem has infinitely many real
eigenvalues if and only if its translation or itself is
$\mathcal{PT}$-symmetric. Also, we will prove some other interesting direct and
inverse spectral results.
