A reality proof in PT -symmetric quantum mechanics

led Bender and Boettcher to conjecture that under suitable boundary conditions the spectrum of (1) was entirely real and positive provided M ≥ 1 [2]. Bender and Boettcher also noticed the relevance of PT -symmetry [2,3] to these problems. For all M , the potentials V (x) = −(ix)2M are invariant under the combined action of parity and time reversal, and this implies the eigenvalues are either real or occur in complex-conjugate pairs. If all energy eigenstates are also eigenstates of the PT operator, then the spectrum is guaranteed to be entirely real, but this is not true in general: the PT -symmetry can be spontaneously broken, in which case the corresponding energies occur in complex-conjugate pairs. In fact, a proof of the reality conjecture has only recently been obtained [4], via a correspondence between ordinary differential equations and integrable models (the so-called