I was going through a paper by Jean Bourgain and it states that an eigenfunction $\psi$ of the Laplacian $\Delta$ with eigenvalue $-\lambda$ on the flat $n-$torus $\mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n$, ie, $\Delta\psi =-\lambda\psi$ can be expanded into the Fourier series \begin{equation*} \psi(x)=\sum_{|\xi|^2=\lambda}\widehat{\psi}(\xi)e^{2\pi i\zeta\cdot x}. \end{equation*} Sorry if this is an obvious question (and consequently something I should know), but what is the reason that an eigenfunction can be expended into such a Fourier series? Does this connection come from the separation of variables of PDEs, or is there another result that clearly states the connection between eigenfunctions and Fourier series expansion?
Thanks in advance for your help and comments.