I have following problem.
Take $A= - f'' + uf $ an unbounded operator with $u \in \mathcal{S}(\mathbb{R},\mathbb{R})$ being the Schwartz class. The domain of $A$ is $H^2(\mathbb{R})$.
(i) Show that A is self-adjoint.
(ii) Show that the continuous spectrum of A is $[0,\infty)$
To show that A is self adjoint I tried using Rellich-Kato and split A into two sections showing that one is self adjoint and the other symmetric. I got stuck trying to show that $B = -f''$ is self adjoint.
For the second part I need to find all the values $\lambda$ such that $A-\lambda I$ is injective and has dense range but is not surjective. I am not sure how to approach this in the best way.
Any input would be greatly appreciated!