Whether the non-compact Riemannian manifold has continuous spectrum ?

Question

As pictures below, the spectrum of compact Riemannian manifold is discrete and the spectrum of $R^2$ or $H$ is continuous. Whether the non-compact Riemannian manifold has continuous spectrum ? How to show it ?

Answer 1

Not every non-compact Riemannian manifold needs to have a continuous spectrum. There are examples of such manifolds which only have a discrete spectrum. See for example this article by A. Baider. On the other hand, there are non-compact Riemannian manifolds with purely continuous spectrum, see here.
The question about the Laplace spectrum for non-compact Riemannian manifolds is quite difficult in general. Already in dimension $2$, for hyperbolic surfaces $\mathbb{H}/\Gamma$ of finite volume one needs the theory of Eisenstein series and their analytic continuation to prove the existence of the continuous spectrum - which is $[\frac{1}{4},\infty)$. Selberg pioneered the study of the spectral theory of hyperbolic surfaces in the 1950’s. He developed his "trace formula". This leads deeply into number theory, the theory of automorphic forms and many other topics.