Consider a bounded Euclidean domain $\Omega \subset \mathbb{R}^n$ (for simplicity, let's say, $\Omega$ has smooth boundary and is simply connected). Let $p \in \Omega$ be a point, and call $\Omega_n = \Omega \setminus B(p, \frac{1}{n})$. Let $\Delta$ (respectively, $\Delta_n$) denote the Dirichlet Laplacian on $\Omega$ (respectively, $\Omega_n$). I am interested in when the eigenvalues and eigenfunctions of $-\Delta_n$ converge to the eigenvalues and eigenfunctions of $-\Delta$, and in what sense (norm etc.) the eigenfunction convergence happens. The same question can be asked about the Neumann Laplacian.

The chief reason for my interest is that I am studying problems in homogenization and inverse problems (for example, papers of Vogelius, Friedman, Ammari et al). In these problems the setting is typically more complex, and problems involve an increasing number of perforations with decreasing "size". I am trying to understand a "toy case" thoroughly.

Edit: The main difficulty in the question seems to be for the case of repeated eigenvalues.

The “hot spots” conjecture for domains with two axes of symmetry, J. Amer. Math. Soc. 13 (2000) 741–772] might be a good source of references. If I find time tomorrow, I will have a look at it. $\endgroup$