## Negative-energy PT-symmetric Hamiltonians

Carl M. Bender, Daniel W. Hook, S. P. Klevansky

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).

http://arxiv.org/abs/1203.6590
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)