Anyonic PT symmetry, drifting potentials and non-Hermitian delocalization

We consider wave dynamics for a Schrodinger equation with a non-Hermitian Hamiltonian H satisfying the generalized (anyonic) parity-time symmetry PT H = exp(2i)HPT , where P and T are the parity and time-reversal operators. For a stationary potential, the anyonic phase just rotates the energy spectrum of H in a complex plane, however, for a drifting potential the energy spectrum is deformed and the scattering and localization properties of the potential show intriguing behaviors arising from the breakdown of the Galilean invariance when ≠= 0. In particular, in the unbroken PT phase the drift makes a scattering potential barrier reflectionless, whereas for a potential well the number of bound states decreases as the drift velocity increases because of a non-Hermitian delocalization transition.