Spectral analysis of the discrete Maxwell operator: The limiting absorption principle

We are interested by the spectral analysis of the anisotropic discrete Maxwell operator ${\widehat{H}}^D$ defined on the square lattice $\mathrm{Z}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\mathrm{Z}}^3$. In aim to prove that the limiting absorption principle holds we construct a conjugate operator to the Fourier series of ${\widehat{H}}^D$ at any not-zero real value. In addition we show that at some particular thresholds the conjugate operator is essentially self-adjoint.