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We present several applications of mode matching methods in spectral andscattering problems. First, we consider the eigenvalue problem for theDirichlet Laplacian in a finite cylindrical domain that is split into twosubdomains by a "perforated" barrier. We prove that the first eigenfunction islocalized in the larger subdomain, i.e., its $L_2$ norm in the smallersubdomain can be made arbitrarily small by setting the diameter of the "holes"in the barrier small enough. This result extends the well known localization ofLaplacian eigenfunctions in dumbbell domains. We also discuss an extension tononcylindrical domains with radial symmetry. Second, we study a scatteringproblem in an infinite cylindrical domain with two identical perforatedbarriers. If the holes are small, there exists a low frequency at which anincident wave is fully transmitted through both barriers. This result iscounter-intuitive as a single barrier with the same holes would fully reflectincident waves with low frequences.

Mode matching methods for spectral and scattering problems