Eigenvalues of PT-symmetric oscillators with polynomial potentials

Abstract

We study the eigenvalue problem $-u^{\prime\prime}(z)-[(iz)^m+P_{m-1}(iz)]u(z)=λu(z)$ with the boundary conditions that $u(z)$ decays to zero as $z$ tends to infinity along the rays $\arg z=-\fracπ{2}\pm \frac{2π}{m+2}$, where $P_{m-1}(z)=a_1 z^{m-1}+a_2 z^{m-2}+...+a_{m-1} z$ is a polynomial and integers $m\geq 3$. We provide an asymptotic expansion of the eigenvalues $λ_n$ as $n\to+\infty$, and prove that for each {\it real} polynomial $P_{m-1}$, the eigenvalues are all real and positive, with only finitely many exceptions.