Anharmonic Oscillators with Infinitely Many Real Eigenvalues and $\mathcal{PT}$-Symmetry

We study the eigenvalue problem $-u''+V(z)u=\lambda u$ in the complex plane with the boundary condition that $u(z)$ decays to zero as $z$ tends to infinity along the two rays $\arg z=-\frac\pi2\pm \frac2\pi{m+2}$, where $V(z)=-(iz)^m-P(iz)$ for complex-valued polynomials $P$ of degree at most $m-1\ge 2$. We provide an asymptotic formula for eigenvalues and a necessary and sufficient condition for the anharmonic oscillator to have infinitely many real eigenvalues.