Eigenvalues of PT-symmetric oscillators with polynomial potentials

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