Let $\Omega\subseteq \mathbb{R}^n\cap \{x_n \geq 0\}$ be a domain with piecewise smooth boundary and suppose that $u\in C^2(\Omega)\cap C^1\left(\bar\Omega\right)$ is an eigenfunction of the Laplacian with Neumann boundary conditions. More precisely, assume that $$\Delta u = \lambda u \quad \text{in } \Omega$$ for some $\lambda\in \mathbb{R}$ and that $\partial_\nu u = 0$ on $\partial\Omega$. Define $$ \tilde{u}(x_1, \dots, x_n) =\begin{cases} u(x_1, \dots, x_n) &\text{if } x_n \geq 0\\ u(x_1, \dots, x_{n-1}, -x_n) &\text{otherwise}. \end{cases} $$ on $\tilde{\Omega} = \Omega \cup \{(x_1, \dots, x_{n-1}, -x_n)\}: (x_1, \dots, x_n)\in \Omega$.
I know that $\tilde{u}$ is an eigenfunction on $\tilde{\Omega}$. Where can I find a reference which proves this fact?
I'm looking for a reference (or proof!) for this widely used reflection method. I appreciate any help!