$\mathbf {\operatorname{SL} _2({\mathbb{R} })}$, exponential Herglotz representations, and spectral averaging

We revisit the concept of spectral averaging and point out its relationship with one-parameter subgroups of $\operatorname{SL}_2({\mathbb{R} })$ and the corresponding Möbius transformations. In particular, we identify exponential Herglotz representations as the basic ingredient for the absolute continuity of averaged spectral measures with respect to Lebesgue measure; the associated spectral shift function turns out to be the corresponding density for the averaged measure. As a by-product of our investigations we unify the treatment of rank-one perturbations of selfadjoint operators and that of selfadjoint extensions of symmetric operators with deficiency indices $(1,1)$. Moreover, we derive separate averaging results for absolutely continuous, singular continuous, and pure point measures and conclude with an averaging result for the $\kappa$-continuous part (with respect to the $\kappa$-dimensional Hausdorff measure) of singular continuous measures.