Nodes of eigenfunctions and Courant's nodal domain theorem I am looking for a reference for properties of eigenfunctions of the Laplacian (on the Euclidean plane, and maybe also Laplace-Beltrami on a general manifold): * The discreteness of the set of eigenvalues, * Nodes of eigenfunctions, * Courant's nodal domain theorem, * The Faber-Krahn inequality, and other related results. I have tried _Methods of Mathematical Physics (Courant, Hilbert)_ but it contains only some of the above, is quite old and a bit hard to read.

I recently read the following survey of various results about the spectrum of the Laplacian (on Euclidean domains, though many results carry over directly to the Laplace-Beltrami operator). You might find it, and its references, helpful: <