Upside-down potentials

A quantum-mechanical theory is PT-symmetric if it is described by a Hamiltonian that commutes with PT, where P represents space reflection and T represents time reversal. Often, but not always, a PT-symmetric Hamiltonian is non-Hermitian, and thus its eigenvalues may not be all real. However, PT-symmetric Hamiltonians are interesting because they usually have a parametric region of unbroken PT symmetry in which the eigenvalues are indeed all real and a region of broken PT symmetry in which the some of the eigenvalues are complex. These regions are separated by a phase transition that has been repeatedly observed in laboratory experiments. This paper focuses on the properties of a particular PT-symmetric Hamiltonian, H = p(2) - x(4), which has an upside-down potential. The spectrum of this PT-symmetric Hamiltonian is rigorously known to be entirely real, and the purpose of this paper is to present an intuitive explanation of why the spectrum of this Hamiltonian is real, positive, and discrete.