Let $\Omega\subset \mathbb{R}^n$ a bounded domain with smooth boundary, $0<\lambda_1\leq \lambda_2 \leq \dots \leq \lambda_k\leq \dots$ the Dirichlet eigenvalues and $\{w_k\}_{k=1}^{+\infty}$ an $L^2(\Omega)$-orthonormal basis made of eigenfunctions ordered by the corresponding eigenvalues. Actually, using rigged Hilbert spaces, $\{w_k\}_{k=1}^{+\infty}$ (up to renormalization) is an orthonormal basis in $H^1(\Omega)$. Hence for any $f\in H^1(\Omega)$ we have $$\sum_{k=1}^{N}a_k w_k(x)\to f(x) \quad \text{in}\quad H^1(\Omega),\quad \text{where}\quad a_k:=\int_{\Omega}w_k(x) f(x) \,dx.$$

Furthermore it is well-known that $w_k(x)\in C^\infty(\bar{\Omega})$ and is real analytical in $\Omega$.

Suppose that $\Omega \subset \mathbb{R}^3$ and $\Omega$ contains the origin.

Main questions.

1. Can we hope that the convergence of $\sum_{k=1}^N a_k w_k(x)$ to $f(x)=\frac{1}{|x|}$ is pointwise in any compact set $K\subset \Omega\setminus \{0\}$? In such case, what can we say about the asymptotic behaviour of the series $S(x)=\sum_{k=1}^{+\infty}a_kw_k(x)$ near $x=0$ (suspecting that $S(x) \sim \frac{1}{|x|}$?
2. If $\Omega=B_1(0)$, using that the eigenfunctions of the Laplacian are the Bessel functions multiplied by spherical armonics, can we assert something more in this special case?

A general problem for $f(x)$ in $\Omega \subset \mathbb{R}^n$: Can we hope that the converge inside $\Omega$ is pointwise where $f(x)$ is continuous? This is sounds like a Carleson theorem for a general (not only rectangular) domain with a non-trigonometric basis. I suppose that this is an open problem. [This is strictly related to this question][1]

Any related idea, counterexample or reference is welcome! [1]: Carleson's Theorem on Manifolds