It is a well known result that for $\Omega$ bounded open set in $\mathbb{R}^n$, there exists a basis of $C^\infty$ eigenfunctions of the Laplacian for $L^2(\Omega)$. It is also known that there exists a basis of $C^\infty$ eigenfunctions of the Laplacian for $L^2(M)$, $M$ being a compact, connected Riemannian manifold. It can be proved with the fundamental solution of the heat equation (see http://www.math.harvard.edu/~canzani/math253/Lecture13.pdf). My question is : can the second result be proved with the first one ? In other words, can we use the eigenfunction basis of the spaces $L^2(U_i)$ ($U_i$ being the open sets in $\mathbb{R}^n$ associated to an atlas on $M$) to build the eigenfunction basis of $L^2(M)$ ? If anybody has an idea or a reference, that would be great.
Thanks !