On the reality of the eigenvalues for a class of PT-symmetric oscillators

We study the eigenvalue problem -u(eta)(z) - [(iz)(m) + P(iz)]u(z) = with the boundary conditions that u(z) decays to zero as z tends to infinity along the rays arg z = -(pi)/(2) +/- (2pi)/(m+2), where P(z) = a(1)z(m-1) + a(2)z(m-2) + ... + a(m-1)z is a real polynomial and m greater than or equal to 2. We prove that if for some 1 less than or equal to j less than or equal to (m)/(2) we have (j - k)a(k) greater than or equal to 0 for all 1 less than or equal to k less than or equal to m - 1, then the eigenvalues are all positive real. We then sharpen this to a larger class of polynomial potentials. In particular, this implies that the eigenvalues are all positive real for the potentials alphaiz(3) + betaz(2) + gammaiz when alpha, beta, gamma is an element of R with alpha not equal 0 and alpha gamma greater than or equal to 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.