PT-symmetric lattices with spatially extended gain/loss are generically unstable

We illustrate, through a series of prototypical examples, that linear parity-time (PT) symmetric lattices with extended gain/loss profiles are generically unstable, for any non-zero value of the gain/loss coefficient. Our examples include a parabolic real potential with a linear imaginary part and the cases of no real and constant or linear imaginary potentials. On the other hand, this instability can be avoided and the spectrum can be real for localized or compact PT-symmetric potentials. The linear lattices are analyzed through discrete Fourier transform techniques complemented by numerical computations.