I am trying to use the spectral theorem for self adjoint operators to decompose the spectrum of the multiplication operator $f(x) = \frac{1}{1+x^2}$ on $L^2(\mathbb{R}).$ This is a problem in Teschl's "Mathematical Applications to Quantum Mechanics." Here is what I have done so far.
The function $f \in L^\infty(\mathbb{R})$ so it is a bounded operator and its spectrum is equal to the closure of the range of $f$ which is the interval $[0,1].$ There are clearly no eigenvectors since $g = \frac{g}{1+x^2}$ implies that $g=0$ a.e. If $\psi(x) \in L^2(\mathbb{R})$ then the spectral measure is defined by $$\mu_\psi(\Omega) = \langle\psi, \chi_{f^{-1}(\Omega)} \psi \rangle$$ for $\Omega \subset \mathbb{R}$ measurable. Since $f$ is smooth and everywhere 2 to 1, if $\Omega$ is a set of Lebesgue measure $0$ then so is $f^{-1}(\Omega)$ so $\mu_\psi$ is absolutely continuous with respect to the Lebesgue measure for all $\psi.$ Therefore the spectrum is entirely absolutely continuous.
I am having trouble finding a spectral basis so that I can decompose the operator into a direct sum of multiplication operators on finite measure spaces. There doesn't seem to be a general procedure for doing this and I can't think of a good place to start.