Exponential decay of Laplacian eigenfunctions in domains with branches of variable cross-sectional profiles

Abstract.We study the behavior of the Laplace operator eigenfunctions in an arbitrary resonator (or waveguide) with branches of variable cross-sectional profiles. When an eigenvalue is below a threshold which is determined by the shape of the branch, the associated eigenfunction is proved to have an upper bound which exponentially decays inside the branch. The decay rate is shown to be twice the square root of the difference between the threshold and the eigenvalue. A finite-element numerical solution of the eigenvalue problem illustrates and further extends the above theoretical result which may help to design elaborate resonators or waveguides in microelectronics, optics and acoustics.

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