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Spectral analysis of a self-similar Sturm-Liouville operator. (English) Zbl 1090.34070
The author considers a family of self-similar Sturm-Liouville operators, indexed by a parameter \(\omega\), and describes the nature of the spectrum (pure point or continuous). This study is motivated by the more general problem of understanding the nature of the spectrum for the class of self-similar Laplacians on finitely ramified self-similar sets. In [Mém. Soc. Math. Fr., Nouv. Sér. 92 (2003; Zbl 1036.82013); in: Michel L. Lapidus (ed.) et al., Fractal geometry and applications: A jubilee of Benoît Mandelbrot. Analysis, number theory, and dynamical systems. In part the proceedings of a special session held during the annual meeting of the American Mathematical Society, San Diego, CA, USA, Januar 2002. Providence, RI: American Mathematical Society (AMS). Proceedings of Symposia in Pure Mathematics 72, Pt. 1, 155–205 (2004; Zbl 1066.37052)], the author proved that the spectral properties of these operators are related to the dynamics of a certain renormalization map, which is a rational map of a smooth complex projective variety. The main question emerging from these investigations is the nature of the spectrum of these operators in the nondegenerate case \(d_\infty=N\), where \(d_\infty\) is the asymptotic degree of the renormalization map, \(N\) is the number of subcells of the set at level 1. The family of self-similar Sturm-Liouville operators considered in the paper corresponds just to the case \(d_\infty=N\).

MSC:
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
34L05 General spectral theory of ordinary differential operators
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