Spectral Analysis of a SelfSimilar SturmLiouville Operator.pdfVIP

Spectral analysis of a self-similar Sturm-Liouville operator Christophe Sabot 4 0 Laboratoire de Probabilit´es et mod`eles al´eatoires, Universit´e Paris 6, 0 2 4, Place Jussieu, 75252 Paris cedex 5, and n a Ecole Normale Sup´erieure, J 1 DMA, 45, rue d’Ulm, 75005 Paris, 0 3 1 v 6 Abstract: In this text we describe the spectral nature (pure point or continuous) 5 0 of a self-similar Sturm-Liouville operator on the line or the half-line. This is motivated 1 by the more general problem of understanding the spectrum of Laplace operators on 0 4 unbounded finitely ramified self-similar sets. In this context, this furnishes the first 0 example of a description of the spectral nature of the operator in the case where the / so-called ”Neumann-Dirichlet” eigenfunctions are absent. h p - h t AMS classification: 34L10(34L20,82B44) a m : v i Key words: Spectral theory, Eingenfunctions expansion, Sturm-Liouville operators, X dynamics in several complex variables, analysis on self-similar sets, fractals. r a 1E-mail address: sabot@ccr.jussieu.fr 1 In this text we consider a family of self-similar Sturm-Liouville operators, indexed by a parameter ω called the blow-up, and describe the nature of the spectrum (pure point or continuous). This study is motivated by the problem of understanding the