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I have come across the following exercise when studying for my upcoming Quantum Mechanics exam.

Let $P:D(P)\to L^2(\mathbb{R})$ be a self-adjoint operator such that for any smooth function $\psi\in D(P)$

$$(P\psi)(x)=-ih\frac{d\psi(x)}{dx}$$

Show that $\sigma(P)=\mathbb{R}$ where $\sigma$ denotes the spectrum.

I have managed to show that the point spectrum $\sigma_p(P)$ is empty and have that the spectrum is given by $\sigma(P)=\sigma_p(P)\cup\sigma_c(P)$, where $\sigma_c(\cdot)$ is the continuous spectrum. In lectures we were told, without proof, that for a self-adjoint operator:

$$\lambda\in \sigma_c(P)\qquad\iff\qquad\exists\text{ a Weyl sequence for }(P,\lambda)$$

My question is: How do I go about finding this sequence?

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  • $\begingroup$ See the answer here. $\endgroup$ – icurays1 Apr 26 '17 at 19:22

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