I'm going to ask 2 questions but they're (I believe) related.
1) Let $z_0$ be an eigenvalue and $\psi$ a corresponding normalized eigenvector. Compute $μ_{\psi}$.
This is the whole question but we could assume operator $A$ which we know has eigenvalue $z_0$ is self-adjoint and $\mu_{\psi}(\Omega)$ is given by $\langle\psi,P_A(\Omega)\psi\rangle$ where $P_A$ is associated PVM to $A$.
2) Show that $z_0$ is an eigenvalue if and only if P({$z_0$})$\neq0$. Show that $Ran(P(\{z_0\}))$is the corresponding eigenspace in this case. Again we could assume $A$ is self-adjoint.
I've already (kind of) answered both of the questions but reason for why i'm asking these questions is both questions seems very intuitive like the moment i saw the first question i immediately said if $\Omega$ contains $z_0$ it is $||\psi||^2$ if not it is $0$. And i would like to answer them more formally \
3) Under the shadow of these 2 questions i would like to ask another question.We know spectrum $$\sigma(A)=\{z\in\mathbb{R} : P_A(z-\epsilon,z+\epsilon)\neq0 \enspace \forall\epsilon>0\}$$ but this does not differentiate between different types of spectrums pure point,point embedded in continuum and purely continuous
.Clearly eigenvalues are in point part of the spectrum and above results shows us $P(\{z_0\})$ is an eigenspace but what happens if $z_0\in$ purely cont.?Could someone give me an example(maybe example from physics like quantum harmonic oscillator etc.)?
Related links Show that eigen-vectors belong in range of projection valued measure. Spectral measure associated to eigenvector of self-adjoint operator
Any hint and solution is appreciated THANKS!
Best Answer
I'm not sure if there's a way to see this directly from the definition of $P_A$, but here is a proof using the resolvent, which is quite natural in view of the proof of the spectral theorem. With $R_A(z)=(A-zI)^{-1}$, we know that $$\langle\psi,R_A(z)\psi\rangle=\langle \psi,\frac{1}{z_0-z}\psi\rangle=\frac{1}{z_0-z}||\psi||^2$$ This is in turn, by the spectral theorem, equal to $F(z):=\int_{\mathbb{R}}\frac{1}{\lambda-z}\,d\mu_{\psi}(\lambda)$, which is the Borel transform of $\mu_\psi$. We may recover a measure from its Borel transform via the Stieljes inversion formula: $$\mu_\psi(\lambda)=\lim_{\delta\downarrow0}\lim_{\epsilon\downarrow 0}\frac{1}{\pi}\int_{-\infty}^{\lambda+\delta}\text{Im}(F(t+i\epsilon))\,dt$$ I suggest your try this computation, but if you cannot get it I am happy to include details. The key point is that the integrand has a singularity as $\epsilon\rightarrow 0$ precisely at $t = z_0$.
If $\psi$ is an eigenvector corresponding to $z_0$ then $$0\neq\langle \psi,\psi\rangle=\int_{\mathbb{R}}\,d\mu_\psi=\int_{\{z_0\}}\,d\mu_\psi=\langle \psi,P(\{z_0\})\psi\rangle$$ where we have used the fact that $\mu_\psi$ is a point-mass from $1.$ Conversely, if $P(\{z_0\})\neq 0$, we may find $\psi$ such that $P(\{z_0\})\psi=\psi$ because $P(\{z_0\})$ is a projection. This also means that $P(\mathbb{R}\setminus\{z_0\})x=0$. Now, the result follows from DisintegratingByParts answer here. They are proving the backwards implication (more or less), but if you look at the argument it works to prove what we want by tracing it in reverse.
The canonical example of an operator with purely ac spectrum is the free energy $-\Delta$, where $\Delta$ is the Laplacian. For this fact, see Theorem 7.8 in Mathematical Methods in Quantum Mechanics by Teschl. Teschl's book is an excellent reference for this material and you may find it free online. While extremely important, this example is a little hard to make sense of because $-\Delta$ is an unbounded operator. For a simpler example, you may try to show that $f(x)\mapsto xf(x)$ on $L^2([0,1])$ has purely ac spectrum.