Normally you would consider eigenvectors only of operators from the same space to itself.
But the Laplacian is usually defined on $C^2$ into $C$, which is a supset of $C^2$. Which leads to my question, in spectral theory, how are eigenvalues of Laplacian usually defined?
Eigenvalues and eigenvectors can be defined in the usual way: $f$ is an eigenvector of $L$ with eigenvalue $\lambda$ if $L f = \lambda f$. In this case $f \in C^2$, and $L f$, which a priori would be in $C$, happens to be in $C^2$ as well. In fact, using elliptic regularity, $f$ is $C^\infty$.