Eur. Phys. J. B (2012) 85: 371
https://doi.org/10.1140/epjb/e2012-30286-8

Regular Article

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

¹ Mathematical Department of the Faculty of Physics, Moscow State University, 119991 Moscow, Russia
² Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS – École Polytechnique, 91128 Palaiseau, France
³ Laboratoire Poncelet (UMI 2615), CNRS – Independent University of Moscow, Bolshoy Vlasyevskiy Pereulok 11, 119002 Moscow, Russia
⁴ Chebyshev Laboratory, Saint Petersburg State University, 14th line of Vasil’evskiy Ostrov 29, Saint Petersburg, Russia

^a e-mail: denis.grebenkov@polytechnique.edu

Received: 3 April 2012
Received in final form: 10 September 2012
Published online: 14 November 2012

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.