I know very well that Laplacian in bounded domain has a discrete spectrum. How about Laplacian in $\mathbb{R}^n$?(not in some fancy-shaped unbounded domain, but the whole domain)
Where can I find such results?
Moreover, is there a counterpart of Hilbert-Schmidt theorem for Laplacian in $\mathbb{R}^n$? Hilbert-Schmidt asserts there is a countable set of eigenfunctions $\phi_n$ so that $x=\sum \langle x,\phi_n\rangle \phi_n,\forall x\in H$.
Is there a similar theorem saying $x=\int_0^\infty \langle x,\phi_\lambda\rangle \phi_\lambda\,\mathrm{d}\lambda,\forall x\in H$ where $\phi_\lambda$ is the eigenfunction of Laplacian to spectral value $\lambda$?