  Consider in $L^2(\R)$ the \Sc operator family $H(g):=-d^2_x+V_g(x)$ depending
on the real parameter $g$, where $V_g(x)$ is a complex-valued but $PT$
symmetric periodic potential. An explicit condition on $V$ is obtained which
ensures that the spectrum of $H(g)$ is purely real and band shaped;
furthermore, a further condition is obtained which ensures that the spectrum
contains at least a pair of complex analytic arcs.
