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Journal of the London Mathematical Society 1999 59(2):620-636; doi:10.1112/S0024610799007243
© 1999 by London Mathematical Society
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© The London Mathematical Society

Eigenfunction Decay and Eigenvalue Accumulation for the Laplacian on Asymptotically Perturbed Waveguides

Julian Edward

Department of Mathematics, Florida International University Miami, FL 33199, USA edwardj{at}fiu.edu

Received 6 January 1997. Revision received 19 June 1997.

An asymptotically perturbed cylinder is a manifold which, to the exterior of a compact set, is of the form RxM, where M is a compact manifold (with or without boundary), and for which the metric approaches the product metric as the axial variable tends to infinity. The properties of eigenfunctions of the Laplacian on asymptotically perturbed cylinders, with either Dirichlet or Neumann boundary conditions if the boundary is non-empty, are studied. For a large class of asymptotic perturbations, the eigenfunctions are proved to decay faster than the reciprocal of any polynomial as the axial variable tends to infinity. The decay estimates are then used to prove that the eigenvalues are of finite multiplicity, and can accumulate at the thresholds only from below. The results are shown to apply to a large class of acoustic and quantum waveguides.


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