Simple Neumann eigenvalues for the Laplace operator in a domain with a small hole

Let ${\mathbb{A}}(\epsilon)$ be the annular domain obtained by removing from a bounded open domain ${\mathbb{I}}^{o}$ of ${\mathbb{R}}^{n}$ a small cavity of size $\epsilon>0$. Then we assume that for some natural index $l$, $\lambda_{l}[{\mathbb{I}}^{o}]>0$ is a simple Neumann eigenvalue of $-\Delta$ in ${\mathbb{I}}^{o}$, and we show that there exists a real valued real analytic function $\hat{\lambda}_{l}(\cdot,\cdot)$ defined in an open neighborhood of $(0,0)$ in ${\mathbb{R}}^{2}$ such that the $l$-th Neumann eigenvalue $\lambda_{l}[{\mathbb{A}}(\epsilon)]$ of $-\Delta$ in ${\mathbb{A}}(\epsilon)$ equals $\hat{\lambda}_{l}(\epsilon,\kappa_{n}\epsilon\log\epsilon)$ and such that $\hat{\lambda}_{l}(0,0)= \lambda_{l}[{\mathbb{I}}^{o}]$. Here $\kappa_{n}=1$ if $n$ is even and $\kappa_{n}=0$ if $n$ is odd. Thus in particular, we show that if $n$ is even $\lambda_{l}[{\mathbb{A}}(\epsilon)]$ can be expanded into a convergent double series of powers of $\epsilon$ and $\epsilon\log\epsilon$ and that if $n$ is odd $\lambda_{l}[{\mathbb{A}}(\epsilon)]$ can be expanded into a convergent series of powers of $\epsilon$. Then related statements have been proved for corresponding eigenfunctions.