A criterion for the reality of the spectrum of PT-symmetric Schrodinger operators with complex-valued periodic potentials

Consider in L-2(R) the Schrodinger operator family H(g) := -d(x)(2) + 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.