In recent years systems or models which are symmetric under combined parity and time-reversal (PT) have met attention, also because the Hamiltonians are not self-adjoint. After recalling some applications we'll focus on a paradigmatic example, known as the imaginary cubic oscillator, this being the Schrödinger operator with potential i x^3.

Though being manifestly not self-adjoint, its spectrum is nevertheless real, as conjectured by Bessis and Zinn-Justin in 1992, later confirmed numerically, and finally proven by several people, e.g. by means of the Bethe ansatz. Here we will give an explanatory proof resting on WKB asymptotics and some standard results in complex analysis (joint work with I. Giordanelli).