Negative-energy PT-symmetric Hamiltonians

The non-Hermitian PT-symmetric quantum-mechanical Hamiltonian $H=p^2+x^2(ix{)}^{ϵ}$ has real, positive, and discrete eigenvalues for all $ϵ\ge 0$. These eigenvalues are analytic continuations of the harmonic-oscillator eigenvalues $E_n=2n+1$ (n=0, 1, 2, 3, …) at $ϵ=0$. 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 $H=p^2+x^2(ix{)}^{ϵ}$ 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, $ϵ$ lies in the range $(4N-6)\mathrm{/}3<ϵ<4N-2$. At the low and high ends of this range, the eigenvalues are all infinite. At the special intermediate value $ϵ=2N-2$ the eigenvalues are the negatives of those of the conventional Hermitian Hamiltonian $H=p^2+x^{2N}$. However, when $ϵ\mathrm{\ne }2N-2$, there are infinitely many complex eigenvalues. Thus, while the positive-spectrum sector of the Hamiltonian $H=p^2+x^2(ix{)}^{ϵ}$ has an unbroken PT symmetry (the eigenvalues are all real), the negative-spectrum sector of $H=p^2+x^2(ix{)}^{ϵ}$ has a broken PT symmetry (only some of the eigenvalues are real).