Negative-energy PT-symmetric Hamiltonians

The non-Hermitian PT-symmetric quantum-mechanical Hamiltonian H = p2 + x2(ix)ε has real, positive and discrete eigenvalues for all ε ⩾ 0. These eigenvalues are analytic continuations of the harmonic-oscillator eigenvalues En = 2n + 1 (n = 0, 1, 2, 3, …) at ε = 0. However, the harmonic oscillator also has negative eigenvalues En = −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 = p2 + x2(ix)ε also has real and 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)/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 = p2 + x2N. However, when ε ≠ 2N − 2, there are infinitely many complex eigenvalues. Thus, while the positive-spectrum sector of the Hamiltonian H = p2 + x2(ix)ε has an unbroken PT symmetry (the eigenvalues are all real), the negative-spectrum sector of H = p2 + x2(ix)ε has a broken PT symmetry (only some of the eigenvalues are real). This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'. (paper)