  This paper demonstrates that complex PT-symmetric periodic potentials possess
real band spectra. However, there are significant qualitative differences in
the band structure for these potentials when compared with conventional real
periodic potentials. For example, while the potentials V(x)=i\sin^{2N+1}(x),
(N=0, 1, 2, ...), have infinitely many gaps, at the band edges there are
periodic wave functions but no antiperiodic wave functions. Numerical analysis
and higher-order WKB techniques are used to establish these results.
