compactly supported eigenfunction

Question

Does it true that there exists a compactly supported eigenfunction corresponding to the first positive eigenvalue $\lambda_1$ of hyperbolic Laplacian operator $\Delta$ on $L^2(S)$, $S$ is a hyperbolic surface? It's not compact but finite volume. Thanks.

Answer 1

The Laplacian is an elliptic operator so the eigenfunctions are real analytic. They cannot be zero on an open subset unless they are zero on the whole (connected) manifold.