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


Authors: Fritz Gesztesy and Konstantin A. Makarov
Original publication: Algebra i Analiz, tom 15 (2003), nomer 3.
Journal: St. Petersburg Math. J. 15 (2004), 393-418
MSC (2000): Primary :, 34B20, 47A11; Secondary :, 34L05, 47A10
DOI: https://doi.org/10.1090/S1061-0022-04-00814-3
Published electronically: March 30, 2004
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Abstract: 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.


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Additional Information

Fritz Gesztesy
Affiliation: Department of Mathematics, University of Missouri, Columbia, MO 65211
Email: fritz@math.missouri.edu

Konstantin A. Makarov
Affiliation: Department of Mathematics, University of Missouri, Columbia, MO 65211
Email: makarov@math.missouri.edu

DOI: https://doi.org/10.1090/S1061-0022-04-00814-3
Keywords: Spectral averaging, $\operatorname{SL}_2({\mathbb{R}})$, M\"obius transformations, Herglotz representations
Published electronically: March 30, 2004
Article copyright: © Copyright 2004 American Mathematical Society

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