Let $\mu_k(\alpha)$ be the $k$th eigenvalue of the Laplacian with Neumann boundary conditions on the annulus $B_1 \setminus \overline{B_\alpha}$, where $B_r \subset \mathbb{R}^d$ denotes the open ball of radius $r>0$ centered at the origin, $d \geq 2$. Let $\tilde{\mu}_k$ be the $k$th eigenvalue of the Laplacian with Neumann boundary conditions on the ball $B_1$.
Is it true that $\mu_k(\alpha) \to \tilde{\mu}_k$ as $\alpha \to 0$ for any $k \in \mathbb{N}$?
I'm quite sure that the answer is yes, but I can find neither a proper reference nor a simple and short argument for that. (I know references which cover the Dirichlet boundary conditions, but not Neumann...)