• Corpus ID: 166228458

Scattering in periodic waveguide: integral representation and spectrum decomposition

@article{Zhang2019ScatteringIP,
  title={Scattering in periodic waveguide: integral representation and spectrum decomposition},
  author={Ruming Zhang},
  journal={arXiv: Analysis of PDEs},
  year={2019}
}

• Ruming Zhang
• Published 27 May 2019
• Mathematics
• arXiv: Analysis of PDEs

Scattering problems in periodic waveguides are interesting but challenging topics in mathematics, both theoretically and numerically. Due to the existence of eigenvalue, the unique solvability of such problems is not always guaranteed. To obtain a unique solution that is physically meaningful, the limiting absorption principle (LAP) is a commenly used method. LAP assumes that the limit of a family of solutions of scattering problems with absorbing material converges, as the absorbing parameter tends to $0$, and the limit is the unique solution that is physically meaningful. It has been proved that LAP holds for periodic waveguides in [Hoa11], and we call the solution obtained from LAP the LAP solution. However, the formulations of LAP solutions from these theoretical results are still very complicated thus it is difficult to apply them to numerical simulations is still very challenging. In this paper, with the help of the Floquet-Bloch theory, we set up a simplified formulation for the LAP solution of scattering problems with an integral representation, when a "small enough" subset is exclueded for the choice of wave numbers. It is expected that an efficient numerical method could be established based on the simplified formulation. Another important topic for scattering problems in periodic waveguides is the spectrum decomposition on periodic operators. As far as we know, there is no such results and we have successfully established the spectrum decomposition of the operator, thus the operator is explicitly described by its eigenvalues and eigenfunctions, which are closely related to the Bloch wave solutions. Then the solution is also decomposed into the generalized eigenfunctions. This gives a better understanding of the structures of the solutions.