Eigenvalues of -symmetric oscillators with polynomial potentials

We study the eigenvalue problem − u ′′( z ) − [(i z ) m + P m −1 (i z )] u ( z ) = λ u ( z ) with the boundary condition that u ( z ) decays to zero as z tends to infinity along the rays in the complex plane, 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 ⩾ 3. We provide an asymptotic expansion of the eigenvalues λ n as n → +∞, and prove that for each real polynomial P m −1 , the eigenvalues are all real and positive, with only finitely many exceptions.