To understand a crucial example in representation theory, I need the explicit spectral decomposition of the differential operator 
$$
Df(x)=(1+x^2)f''(x)+2xf'(x)
$$ on $L^2({\mathbb R})$. I'm not an expert, but at first glance, theory tells me the existence of a spectral measure, but not what it looks like. Is the spectral measure absolutely continuous with respect to the Lebesgue measure? What are the multiplicities? Has it been explicitly computed somewhere? 



Spectral decomposition of a specific operator
This is a Sturm-Liouville operator $(Df)(x)=(pf')'$, with $p=1+x^2$. These can be rewritten as Schrodinger equations, by using what I would call a Kummer-Liouville transformation. By some conspiracy, all reference to these on the internet seems to have disappeared, but see perhaps my answer [here.](https://mathoverflow.net/questions/204538/uniform-boundedness-of-eigenfunctions-of-an-elementary-differential-operator/204606#204606)


We introduce the new variables $t=\int\_0^x p^{-1/2}(s)\, ds$, $u=p^{1/4}f$, and then $-D$ becomes the Schrodinger operator
$$
Lu = - \frac{d^2u}{dt^2} + V(t)u , \quad V= \frac{p''}{4} - \frac{p'^2}{16p} .
$$
That gives
$$
V = \frac{1}{4} - \frac{1}{4(1+x^2)} = \frac{1}{4} + O( e^{-|t|}) .
$$
Since this is almost constant, with a rapidly decaying error term, standard results show that we have purely absolutely continuous spectrum of multiplicity $2$ on $\sigma\_{ac}(-D)=[1/4, \infty)$, and some (non-empty, finite) discrete spectrum below $1/4$.


