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Continuity of spectral averaging


Author: C. A. Marx
Journal: Proc. Amer. Math. Soc. 139 (2011), 283-291
MSC (2010): Primary 81Q10, 81Q15, 47B15; Secondary 47B36, 47B80
DOI: https://doi.org/10.1090/S0002-9939-2010-10629-9
Published electronically: August 20, 2010
MathSciNet review: 2729090
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider averages $ \kappa$ of spectral measures of rank one perturbations with respect to a $ \sigma$-finite measure $ \nu$. It is examined how various degrees of continuity of $ \nu$ with respect to $ \alpha$-dimensional Hausdorff measures ( $ 0 \leq \alpha \leq 1$) are inherited by $ \kappa$. This extends Kotani's trick where $ \nu$ is simply the Lebesgue measure.


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  • 1. B. Simon, Trace ideals and their applications, 2nd edition, Amer. Math. Soc., Providence, RI, 2005. MR 2154153 (2006f:47086)
  • 2. S. Kotani, Lyapunov exponents and spectra for one-dimensional random Schrödinger operators, Contemporary Math., vol. 50, Amer. Math. Soc., Providence, RI, 1984, pp. 277-286. MR 841099 (88a:60116)
  • 3. V. A. Javrjan, A certain inverse problem for Sturm-Liouville operators, Izv. Akad. Nauk Armjan. SSR Ser. Mat. 6, 1971, pp. 246-251 [Russian]. MR 0301565 (46:723)
  • 4. R. del Rio, C. Martinez, H. Schulz-Baldes, Spectral averaging techniques for Jacobi matrices, J. Math. Phys. 49, 2008, no. 2, 023507, 13 pp. MR 2392867 (2009a:47065)
  • 5. F. Gesztesy and K. A. Makarov, SL(2, $ \mathbb{C}$), exponential Herglotz representations, and spectral averaging, S. Petersburg Math. J. 15, 2004, pp. 393-418. MR 2052165 (2006f:47004)
  • 6. V. A. Javrjan, On the regularized trace of the difference between two singular Sturm-Liouville operators, Sov. Math. Dokl. 7, 1966, pp. 888-891.
  • 7. B. Simon, Spectral averaging and the Krein spectral shift, Proc. Amer. Math. Soc., 126, 1998, pp. 1409-1413. MR 1443857 (98j:47030)
  • 8. R. Carmona, One-dimensional Schrödinger operators with random or deterministic potentials: New spectral types, J. Func. Anal. 51, 1983, pp. 229-258. MR 701057 (85k:34144)
  • 9. B. Simon and T. Wolff, Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians, Comm. Pure Appl. Math. 39, 1986, pp. 75-90. MR 820340 (87k:47032)
  • 10. R. Del Rio, S. Jitomirskaya, Y. Last and B. Simon, Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank one perturbations, and localization, J. d'Analyse Math. 69, 1996, pp. 153-200. MR 1428099 (97m:47002)
  • 11. P. W. Jones and A. G. Poltoratski, Asymptotic growth of Cauchy transforms, Ann. Acad. Sci. Fenn. Math. 29, no. 1, 2004, pp. 99-120. MR 2041701 (2005d:30061)
  • 12. V. Jakšić and Y. Last, A new proof of Poltoratskii's theorem, J. Funct. Anal. 215, 2004, pp. 103-110. MR 2085111 (2005d:47027)
  • 13. C. A. Rodgers, Hausdorff Measures, Cambridge Univ. Press, London, 1970. MR 0281862 (43:7576)
  • 14. C. A. Rodgers and S. J. Taylor, The analysis of additive set functions in Euclidean space, Acta Math. 101, 1959, pp. 273-302. MR 0107690 (21:6413)
  • 15. C. A. Rodgers and S. J. Taylor, Additive set functions in Euclidean space. II, Acta Math. 109, 1963, pp. 207-240. MR 0160860 (28:4070)
  • 16. Y. Last, Quantum dynamics and decompositions of singular continuous spectra, J. Funct. Anal. 142, 1996, no. 2, pp. 406-445. MR 1423040 (97k:81044)

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Additional Information

C. A. Marx
Affiliation: Department of Mathematics, University of California, Irvine, California 92717
Email: cmarx@math.uci.edu

DOI: https://doi.org/10.1090/S0002-9939-2010-10629-9
Received by editor(s): September 21, 2009
Published electronically: August 20, 2010
Additional Notes: The author was supported by NSF Grant DMS - 0601081.
Communicated by: Walter Craig
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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