On the sharp effect of attaching a thin handle on the spectral rate of convergence

TL;DR

This paper investigates how attaching a thin handle to a domain affects the spectral convergence of the Dirichlet Laplacian, providing sharp estimates and analyzing eigenvalue splitting and bifurcation phenomena.

Contribution

It offers explicit nondegeneracy conditions and sharp control of eigenvalue and eigenfunction convergence rates under domain perturbations involving thin handles.

Findings

Eigenvalues and eigenfunctions vary continuously with the channel size.

Sharp convergence rates are established under explicit conditions.

Resonant domains exhibit polynomial eigenvalue splitting and bifurcation.

Abstract

Consider two domains connected by a thin tube: it can be shown that the resolvent of the Dirichlet Laplacian is continuous with respect to the channel section parameter. This in particular implies the continuity of isolated simple eigenvalues and the corresponding eigenfunctions with respect to domain perturbation. Under an explicit nondegeneracy condition, we improve this information providing a sharp control of the rate of convergence of the eigenvalues and eigenfunctions in the perturbed domain to the relative eigenvalue and eigenfunction in the limit domain. As an application, we prove that, again under an explicit nondegeneracy con- dition, the case of resonant domains features polinomial splitting of the two eigenvalues and a clear bifurcation of eigenfunctions.