Negative-energy PT-symmetric Hamiltonians

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... Show more