  The non-Hermitian PT-symmetric quantum-mechanical Hamiltonian
$H=p^2+x^2(ix)^\epsilon$ has real, positive, and discrete eigenvalues for all
$\epsilon\geq 0$. These eigenvalues are analytic continuations of the
harmonic-oscillator eigenvalues $E_n=2n+1$ (n=0, 1, 2, 3, ...) at $\epsilon=0$.
However, the harmonic oscillator also has negative eigenvalues $E_n=-2n-1$
(n=0, 1, 2, 3, ...), and one may ask whether it is equally possible to continue
analytically from these eigenvalues. It is shown in this paper that for
appropriate PT-symmetric boundary conditions the Hamiltonian
$H=p^2+x^2(ix)^\epsilon$ also has real and {\it negative} discrete eigenvalues.
The negative eigenvalues fall into classes labeled by the integer N (N=1, 2, 3,
...). For the Nth class of eigenvalues, $\epsilon$ lies in the range
$(4N-6)/3<\epsilon<4N-2$. At the low and high ends of this range, the
eigenvalues are all infinite. At the special intermediate value $\epsilon=2N-2$
the eigenvalues are the negatives of those of the conventional Hermitian
Hamiltonian $H=p^2+x^{2N}$. However, when $\epsilon\neq 2N-2$, there are
infinitely many complex eigenvalues. Thus, while the positive-spectrum sector
of the Hamiltonian $H=p^2+x^2(ix)^\epsilon$ has an unbroken PT symmetry (the
eigenvalues are all real), the negative-spectrum sector of
$H=p^2+x^2(ix)^\epsilon$ has a broken PT symmetry (only some of the eigenvalues
are real).
