Fine dimensional properties of spectral measures

  • Michael Landrigan

    Catalina Holdings, San Diego, USA
  • Matthew Powell

    University of California, Irvine, USA
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Operators with zero dimensional spectral measures appear naturally in the theory of ergodic Schrödinger operators. We develop the concept of a complete family of Hausdorff measure functions in order to analyze and distinguish between these measures with any desired precision.We prove that the dimension of spectral measures of half-line operators with positive upper Lyapunov exponent are at most logarithmic for every possible boundary phase. We show that this is sharp by constructing an explicit operatorwhose spectralmeasure obtains this dimension. We also extend and improve some basic results from the theory of rank one perturbations and quantum dynamics to encompass generalized Hausdorff dimensions.

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Michael Landrigan, Matthew Powell, Fine dimensional properties of spectral measures. J. Spectr. Theory 12 (2022), no. 3, pp. 1255–1293

DOI 10.4171/JST/436