The potential (iz)^m generates real eigenvalues only, under symmetric   rapid decay conditions
  We consider the eigenvalue problems -u"(z) +/- (iz)^m u(z) = lambda u(z), m >= 3, under every rapid decay boundary condition that is symmetric with respect to the imaginary axis in the complex z-plane. We prove that the eigenvalues lambda are all positive real. 
K.C. Shin