  The PT-symmetric Hamiltonian $H=p^2+x^2(ix)^\epsilon$ ($\epsilon$ real)
exhibits a phase transition at $\epsilon=0$. When $\epsilon\geq0$, the
eigenvalues are all real, positive, discrete, and grow as $\epsilon$ increases.
However, when $\epsilon<0$ there are only a finite number of real eigenvalues.
As $\epsilon$ approaches -1 from above, the number of real eigenvalues
decreases to one, and this eigenvalue becomes infinite at $\epsilon=-1$. 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 $H^{(2n)}=p^{2n}+x^2(ix)^\epsilon$ ($\epsilon$ real, n=1, 2, 3,
...). The complex classical behaviors of these Hamiltonians are also examined.
