TITLE: What is spectrum for Laplacian in $\mathbb{R}^n$?
QUESTION [31 upvotes]: 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 $\lambda$?

REPLY [6 votes]: Hilbert-Schmidt operators are compact. The resolvent $(\Delta-\lambda I)^{-1}$ is not compact and $\Delta$ is not compact. Otherwise you would end up discrete spectrum, which you do not have.
The spectral resolution of the identity for $-\Delta$ is not as simple as $f=\int_{0}^{\infty}(f,\phi_{\lambda})\phi_{\lambda}d\lambda$ because there are so many approximate eigenfunctions with eigenvalue $r > 0$ that they are indexed by a vector $\vec{\xi}$ on a sphere of radius $\sqrt{r}$:
$$
   -\Delta(e^{i\vec{\xi}\cdot\vec{x}})
      =|\vec{\xi}|^{2}e^{i\vec{\xi}\cdot \vec{x}}
$$
Every $\phi_{\vec{\xi}}(\vec{x})=e^{i\vec{\xi}\cdot\vec{x}}$ for $|\vec{\xi}|=\sqrt{r}$ is a classical solution of $-\Delta\phi_{\vec{\xi}} = r\phi_{\vec{\xi}}$. The expansion that you want is obtained from the Fourier transform by writing the inversion integral as an outer integral over a radius, and an inner integral over the surface of a sphere of radius $r$:
$$
                   f(\vec{x})=\frac{1}{(2\pi)^{n/2}}\int_{0}^{\infty}\left(\int_{|\vec{\xi}|=r}e^{i\vec{\xi}\cdot\vec{x}}\hat{f}(\vec{\xi})dS(\vec{\xi})\right)dr,
$$
where $dS$ is the surface measure on the spherical shell. Classically speaking (not necessarily in $L^{2}(\mathbb{R}^{n})$):
$$
               -\Delta \left(\int_{|\vec{\xi}|=r}e^{i\vec{\xi}\cdot\vec{x}}\hat{f}(\vec{\xi})dS(\vec{\xi})\right) = r^{2}\left(\int_{|\vec{\xi}|=r}e^{i\vec{\xi}\cdot\vec{x}}\hat{f}(\vec{\xi})dS(\vec{\xi})\right)
$$
The spectrum of $-\Delta$ is $[0,\infty)$ and it is all continuous spectrum. The spectral measure $E$ for $-\Delta$ is given on any closed interval $[a,b]\subset [0,\infty)$ by the expression
$$
    E[a,b]f = (\chi_{\sqrt{a} \le|\vec{\xi}|\le \sqrt{b}}\hat{f}(\vec{\xi}))^{\wedge}.
$$
The spectral theorem for $-\Delta$ is
$$
                -\Delta f = \int_{0}^{\infty} \lambda dE(\lambda)f,\;\;\; f\in\mathcal{D}(-\Delta).
$$