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 a PT-symmetric periodic optical potential. We prove that the second critical point, 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 it 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 k-th critical points after which the real parts of the (2k-3)th and (2k-2)th bands disappear, where k=3,4,...