On the Reality of the Eigenvalues for a Class of Pt -symmetric Oscillators

We study the eigenvalue problem −u′′(z) − [(iz) + P (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 = − 2 ± 2π m+2 , where P (z) = a1z m−1 + a2z m−2 + · · ·+ am−1z is a real polynomial and m ≥ 2. We prove that if for some 1 ≤ j ≤ m 2 , we have (j − k)ak ≥ 0 for all 1 ≤ k ≤ m− 1, then the eigenvalues are all positive real. We then sharpen this to a slightly larger class of polynomial potentials. In particular, this implies that the eigenvalues are all positive real for the potentials αiz + βz + γiz when α, β, γ ∈ R with α 6= 0 and αγ ≥ 0, and with the boundary conditions that u(z) decays to zero as z tends to infinity along the positive and negative real axes. This verifies a conjecture of Bessis and Zinn-Justin.