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Mode matching methods for spectral and scattering problems. (arXiv:1711.03277v1 [math-ph])
来源于:arXiv
We present several applications of mode matching methods in spectral and
scattering problems. First, we consider the eigenvalue problem for the
Dirichlet Laplacian in a finite cylindrical domain that is split into two
subdomains by a "perforated" barrier. We prove that the first eigenfunction is
localized in the larger subdomain, i.e., its $L_2$ norm in the smaller
subdomain can be made arbitrarily small by setting the diameter of the "holes"
in the barrier small enough. This result extends the well known localization of
Laplacian eigenfunctions in dumbbell domains. We also discuss an extension to
noncylindrical domains with radial symmetry. Second, we study a scattering
problem in an infinite cylindrical domain with two identical perforated
barriers. If the holes are small, there exists a low frequency at which an
incident wave is fully transmitted through both barriers. This result is
counter-intuitive as a single barrier with the same holes would fully reflect
incident waves with lo 查看全文>>