Exact artificial boundary conditions for problems with periodic structures

Based on the work of Zheng on the artificial boundary condition for the Schrِdinger equation with sinusoidal potentials at infinity, an analytical impedance expression is presented for general second-order ODE problems with periodic coefficients and its validity is shown to be strongly supported by numerical evidences. This new expression for the kernel of the Dirichlet-to-Neumann mapping of the artificial boundary conditions is then used for computing the bound states of the Schrِdinger operator with periodic potentials at infinity. Other potential applications are associated with the exact artificial boundary conditions for some time-dependent problems with periodic structures. As an example, a two-dimensional hyperbolic equation modeling the TM polarization of the electromagnetic field with a periodic dielectric permittivity is considered.