Eigenvalues of PT-symmetric oscillators with polynomial potentials

We study the eigenvalue problem -u''(z) - [(iz)m + Pm-1(iz)]u(z) λu(z) with the boundary condition that u(z) decays to zero as z tends to infinity along the rays arg z = -π/2 ± 2π/(m+2) in the complex plane, where Pm-1(z) = a1zm-1 + a2zm-2 + . . . + am-1z is a polynomial and integers m ≥ 3. We provide an asymptotic expansion of the eigenvalues λn as n → +∞, and prove that for each real polynomial Pm-1, the eigenvalues are all real and positive, with only finitely many exceptions