Continuity of spectral averaging

Marx, C.

doi:10.1090/S0002-9939-2010-10629-9

Continuity of spectral averaging
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by C. A. Marx PDF

Proc. Amer. Math. Soc. 139 (2011), 283-291 Request permission

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.

References

Barry Simon, Trace ideals and their applications, 2nd ed., Mathematical Surveys and Monographs, vol. 120, American Mathematical Society, Providence, RI, 2005. MR 2154153, DOI 10.1090/surv/120
S. Kotani, Lyapunov exponents and spectra for one-dimensional random Schrödinger operators, Random matrices and their applications (Brunswick, Maine, 1984) Contemp. Math., vol. 50, Amer. Math. Soc., Providence, RI, 1986, pp. 277–286. MR 841099, DOI 10.1090/conm/050/841099
V. A. Javrjan, A certain inverse problem for Sturm-Liouville operators, Izv. Akad. Nauk Armjan. SSR Ser. Mat. 6 (1971), no. 2–3, 246–251 (Russian, with Armenian and English summaries). MR 0301565
Rafael del Rio, Carmen Martinez, and Hermann Schulz-Baldes, Spectral averaging techniques for Jacobi matrices, J. Math. Phys. 49 (2008), no. 2, 023507, 13. MR 2392867, DOI 10.1063/1.2840910
F. Gestezi and K. A. Makarov, $\textrm {SL}_2(\textbf {R})$, exponential representation of Herglotz functions, and spectral averaging, Algebra i Analiz 15 (2003), no. 3, 104–144 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 15 (2004), no. 3, 393–418. MR 2052165, DOI 10.1090/S1061-0022-04-00814-3
V. A. Javrjan, On the regularized trace of the difference between two singular Sturm-Liouville operators, Sov. Math. Dokl. 7, 1966, pp. 888-891.
Barry Simon, Spectral averaging and the Krein spectral shift, Proc. Amer. Math. Soc. 126 (1998), no. 5, 1409–1413. MR 1443857, DOI 10.1090/S0002-9939-98-04261-0
René Carmona, One-dimensional Schrödinger operators with random or deterministic potentials: new spectral types, J. Funct. Anal. 51 (1983), no. 2, 229–258. MR 701057, DOI 10.1016/0022-1236(83)90027-7
Barry Simon and Tom Wolff, Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians, Comm. Pure Appl. Math. 39 (1986), no. 1, 75–90. MR 820340, DOI 10.1002/cpa.3160390105
R. del Rio, S. Jitomirskaya, Y. Last, and B. Simon, Operators with singular continuous spectrum. IV. Hausdorff dimensions, rank one perturbations, and localization, J. Anal. Math. 69 (1996), 153–200. MR 1428099, DOI 10.1007/BF02787106
P. W. Jones and A. G. Poltoratski, Asymptotic growth of Cauchy transforms, Ann. Acad. Sci. Fenn. Math. 29 (2004), no. 1, 99–120. MR 2041701
Vojkan Jakšić and Yoram Last, A new proof of Poltoratskii’s theorem, J. Funct. Anal. 215 (2004), no. 1, 103–110. MR 2085111, DOI 10.1016/j.jfa.2003.09.014
C. A. Rogers, Hausdorff measures, Cambridge University Press, London-New York, 1970. MR 0281862
C. A. Rogers and S. J. Taylor, The analysis of additive set functions in Euclidean space, Acta Math. 101 (1959), 273–302. MR 107690, DOI 10.1007/BF02559557
C. A. Rogers and S. J. Taylor, Additive set functions in Euclidean space. II, Acta Math. 109 (1963), 207–240. MR 160860, DOI 10.1007/BF02391813
Yoram Last, Quantum dynamics and decompositions of singular continuous spectra, J. Funct. Anal. 142 (1996), no. 2, 406–445. MR 1423040, DOI 10.1006/jfan.1996.0155