spectral theory of Laplacian on $\mathbb R^n$

Question

Can you describe the spectrum of the Laplacian $ \Delta : H^2(\mathbb R^n) \subset L^2(\mathbb R^n) \rightarrow L^2(\mathbb R^n)$?

I am interested for which values $z \in \mathbb C$ the equation

$\Delta u + z u = f$

has a solution $u \in H^2(\mathbb R^n)$ for every $f \in L^2(\mathbb R^n)$. I think that the point spectrum is empty, but is every $z$ in the resolvent set?

Answer 1

You should use Fourier transform.

Taking the Fourier transform we see that is equivalent to $$(-\lvert\xi\rvert^2 +z)\hat{u}=\hat{g}, $$ which has the unique formal solution $$\tag{2}\hat{u}=-\frac{\hat{g}}{-z+ \lvert\xi\rvert^2}.$$

Now you have to prove that this formal solution make sense in $H^2$ if and only if $z\leq 0$.