I am stuck with the following problem:
Given the momentum operator of quantum mechanics $Af = - if'$ where the domain $D(A)$ consists of functions $f \in L^2(\mathbb{R})$ that are absolutely continuous and such that $f' \in L^2(\mathbb{R})$.
(i) Prove that $A$ has no eigenvalues.
(ii) Prove that the continuous spectrum of $A$ is $\mathbb{R}$.
We have already shown in class that $A$ is self-adjoint.
For (i) I would use the equation for eigenvalues $Af = \lambda f$ and then show that this cannot hold. But I came up with a solution which shouldn't be the case.
For (ii) we have to construct a sequence $\{f_n\}_{n\geq1}$ in $D(A)$ with $||f_n||=1$ and $||(A-\lambda I)f_n|| \to 0$ for all $\lambda \in \mathbb{R}$, but I have problems finding this sequence.
I would be grateful for any input.