 02191 Fritz Gesztesy and Konstantin A. Makarov
 $\mathbf {SL_2(\bbR)}$, Exponential Herglotz Representations, and Spectral Averaging
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Apr 22, 02

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Abstract. We revisit the concept of spectral averaging and point out its origin in
connection with oneparameter subgroups of $SL_2(\bbR)$ and
the corresponding M\"obius transformations. In particular, we identify
exponential Herglotz representations as the basic ingredient for the
absolute continuity of average spectral measures with respect to
Lebesgue measure and the associated spectral shift function as the
corresponding density for the averaged measure. As a byproduct of our
investigations we unify the treatment of rankone 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, singularly continuous, and
pure point measures and conclude with an averaging result of the
$\kappa$continuous part (with respect to the $\kappa$dimensional
Hausdorff measure) of singularly continuous measures.
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