  We introduce the one-dimensional PT-symmetric Schrodinger equation, with
complex potentials in the form of the canonical superoscillatory and
suboscillatory functions known in quantum mechanics and optics. While the
suboscillatory-like potential always generates an entirely real eigenvalue
spectrum, its counterpart based on the superoscillatory wave function gives
rise to an intricate pattern of PT-symmetry-breaking transitions, controlled by
the parameters of the superoscillatory function. One scenario of the
transitions proceeds smoothly via a set of threshold values, while another one
exhibits a sudden jump to the broken PT symmetry. Another noteworthy finding is
the possibility of restoration of the PT symmetry, following its original loss,
in the course of the variation of the parameters.
