Eigenvalues of PT-symmetric oscillators with polynomial potentials

Abstract

We study the eigenvalue problem $-u^{\prime \prime }(z)-[(iz{)}^m+P_{m-1}(iz)]u(z)=\lambda u(z)$ 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 $P_{m-1}(z)=a_1{z}^{m-1}+a_2{z}^{m-2}+...+a_{m-1}z$ is a polynomial and integers $m\ge 3$. We provide an asymptotic expansion of the eigenvalues ${\lambda }_n$ 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.

23 pages, 1 figure. v2: equation (14) as well as a few subsequent equations has been changed. v3: typos corrected

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