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On a spectral problem related to self-similar measures. (English) Zbl 0823.34071

Let \(\mu\) be a self-similar finite “fractal” measure on (0,1), see [J. E. Hutchinson, Indiana Univ. Math. J. 30, 713-747 (1981; Zbl 0598.28011)] for the description of this class of measures. The quadratic form \(\int^ 1_ 0 | x(t) |^ 2d \mu(t)\) is bounded in the Sobolev space \(H_ 0^ 1 (0,1)\) and generates there a selfadjoint and compact operator. The behavior of its eigenvalue distribution function \(n(\lambda)\) at \(\lambda = 0\) is investigated. A number \(\delta \in (0,{1 \over 2})\) is associated with \(\mu\), such that \(n(\lambda) = O(\lambda^{- \delta})\). Moreover, it is shown that \(\lambda^ \delta (\lambda)\) either tends to a nonzero limit, or behaves as a periodic function of \(\log \lambda\) as \(\lambda \to 0\). These two cases are distinguished by certain arithmetic properties of \(\mu\). Relations between the Hausdorff dimension of \(\mu\), that of its closed support, and the number \(\delta\), are also discussed.

MSC:

34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
47A75 Eigenvalue problems for linear operators
28A80 Fractals

Citations:

Zbl 0598.28011
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