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

Anharmonic oscillators in the complex plane, $\mathcal{PT}$-symmetry, and real eigenvalues