Let $\Omega \subset \mathbb R^n$ be a bounded domain with a smooth boundary that contains the origin. Let us consider the following classical linear wave equation $$ \begin{cases} \partial^2_t u -\Delta u = 0 \quad \text{on $(0,\infty)\times \Omega$}\\ u(t,x)=0 \quad \text{for $(t,x)\in (0,\infty)\times \partial\Omega$} \\ u(0,x)=\delta(x), \partial_t u(0,x)=0 \quad \text{for $x\in \Omega$}. \end{cases} $$ It is classical that the above initial boundary value problem admits a unique solution $$ u \in C([0,T]; H^{-\frac{n}{2}-\epsilon}(\Omega)) \cap C^1([0,T]; H^{-\frac{n}{2}-1-\epsilon}(\Omega))$$ for any $T>0$ and $\epsilon>0$.
My question is whether there is a way to represent this solution using the Dirichlet eigenfunctions of $-\Delta$ on $\Omega$. In other words is there any way to make sense of the "formal solution": $$ u(t,x) = \sum_{k=1}^{\infty} \phi_k(0)\, \cos(\sqrt{\lambda_k}t) \phi_k(x)?$$
