The spectrum of the Hamiltonian with a PT-symmetric Periodic Optical Potential
KünyeVeliev, O. A. (2017). The Spectrum of the Hamiltonian with a PT-symmetric Periodic Optical Potential. International Journal of Geometric Methods in Modern Physics, 15(1), 1-25. http://dx.doi.org/10.1142/S0219887818500081
We give a complete description, provided with a mathematical proof, of the shape of the spectrum of the Hill operator with potential 4 cos2 x+4iV sin 2x, where V ∈ (0, ∞). We prove that the second critical point V2, after which the real parts of the first and second band disappear, is a number between 0.8884370025 and 0.8884370117. Moreover we prove that V2 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, ...