  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.
