Universal spectral behavior of $x^2(ix{)}^{ϵ}$ potentials

The PT-symmetric Hamiltonian $H=p^2+x^2(ix{)}^{ϵ}$ ($ϵ$ real) exhibits a phase transition at $ϵ=0$. When $ϵ\ge 0$, the eigenvalues are all real, positive, discrete, and grow as $ϵ$ increases. However, when $ϵ<0$ there are only a finite number of real eigenvalues. As $ϵ$ approaches -1 from above, the number of real eigenvalues decreases to one, and this eigenvalue becomes infinite at $ϵ=-1$. In this paper it is shown that these qualitative spectral behaviors are generic and that they are exhibited by the eigenvalues of the general class of Hamiltonians $H^{(2n)}=p^{2n}+x^2(ix{)}^{ϵ}$ ($ϵ$ real, n=1, 2, 3, …). The complex classical behaviors of these Hamiltonians are also examined.