I am interested in the Laplace operator $\Delta$ on a concave polygon. When the polygon is convex, it is known that $\Delta: H^2(\Omega) \rightarrow L^2(\Omega)$ is boundedly invertible. In addition, when there is an obtuse angle a similar result holds for $\Delta: H_w^2(\Omega)\rightarrow L_w^2(\Omega)$, where $H_w^2, L_w^2$ are weighted Sobolev spaces. In this case, the weight function depends on the distance from the corners, $L^2$ is continuously embedded in $L^2_w$, and the Laplacian is again isomorphism. For a special choice of weights (depending on the size of the angles in the polygon), $A^{-1}: L_w^2(\Omega)\rightarrow L_w^2(\Omega)$ is a compact operator. Now, I wonder is it possible to have an orthonormal basis for $L_w^2(\Omega)$ of eigenfunctions for $\Delta$ when $\Omega$ is a concave polygon?