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It is shown that the PT symmetric Hamiltonian with the periodic potentialV(x) = 4 cos^2 x + 4 i V_0 sin 2x can be mapped into a Hermitian Hamiltonianfor $V_0<0.5$, by a similarity transformation. It is also shown that thereexist a second critical point of the potential V(x), apart from the knowncritical point $V_0=0.5$, for $V_0^c ~ .888437$ after which no part of theeigenvalues and the band structure remains real. Relevant physical consequenceof this finding has been pointed out.

A note on the PT-invariant periodic potential V(x)=4 cos^2 x + 4 i V_0 sin 2x