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Exact artificial boundary conditions for problems with periodic structures

Ehrhardt, Matthias; Chunxiong, Zheng

Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin

Based on the work of Zheng on the artificial boundary condition for the Schrödinger equation with sinusoidal potentials at infinity, an analytical impedance expression is presented for general second order periodic ODE problems and its validity is shown to be strongly supported by numerical evidences. This new expression for the kernel of artificial boundary conditions is then used for computing the bound states of the Schrödinger 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 TE polarization of the electromagnetic field with a periodic dielectric permittivity is considered.