Exponential decay of Laplacian eigenfunctions in domains with branches

TL;DR

This paper proves that Laplacian eigenfunctions decay exponentially in branched domains when eigenvalues are below a certain threshold, with the decay rate explicitly related to the eigenvalue and domain shape.

Contribution

It provides a general exponential decay estimate for Laplacian eigenfunctions in domains with branches, applicable in any dimension and supported by numerical simulations.

Findings

Eigenfunctions decay exponentially in branches when eigenvalues are below a threshold.

Decay rate is twice the square root of the difference between the threshold and the eigenvalue.

Numerical simulations confirm and extend the theoretical decay estimates.

Abstract

The behavior of Laplacian eigenfunctions in domains with branches is investigated. If an eigenvalue is below a threshold which is determined by the shape of the branch, the associated eigenfunction is proved to exponentially decay inside the branch. The decay rate is twice the square root of the difference between the threshold and the eigenvalue. The derived exponential estimate is applicable for arbitrary domains in any spatial dimension. Numerical simulations illustrate and further extend the theoretical estimate.