The spectrum of the Hamiltonian with a PT-symmetric periodic optical potential

We give a complete description, provided with a mathematical proof, of the shape of the spectrum of the Hill operator with potential 4 cos(2) x+ 4iV sin 2x, where V is an element of (0,infinity). We prove that the second critical point V-2, after which the real parts of the first and second bands disappear, is a number between 0.8884370025 and 0.8884370117. Moreover, we prove that V-2 is the degeneration point for the first periodic eigenvalue. Besides, we give a scheme by which one can find arbitrary precise value of the second critical point as well as the kth critical points after which the real parts of the (2k - 3) th and (2k - 2) th bands disappear, where k = 3, 4,....