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

For integers $m\ge 3$ and $1\le \mathrm{\ell }\le m-1$, we study the eigenvalue problems $-u^{\mathrm{\prime }\mathrm{\prime }}(z)+[(-1{)}^{\mathrm{\ell }}(iz{)}^m-P(iz)]u(z)=\lambda u(z)$ with the boundary conditions that $u(z)$ decays to zero as $z$ tends to infinity along the rays $\arg z=-\frac{\pi }{2}\pm \frac{(\mathrm{\ell }+1)\pi }{m+2}$ 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 $\mathcal{P}\mathcal{T}$-symmetric, then the eigenvalues are all real and positive with at most finitely many exceptions. Moreover, we show that when $\gcd (m,\mathrm{\ell })=1$, the eigenvalue problem has infinitely many real eigenvalues if and only if its translation or itself is $\mathcal{P}\mathcal{T}$-symmetric. Also, we will prove some other interesting direct and inverse spectral results.