Spectral analysis of a self-similar Sturm-Liouville operator Christophe Sabot 34L1034L2082B44spectral theoryEingenfunctions expansionSturm-Liouville operatorsdynamics in several complex variablesanalysis on self-similar setsfractals In this text we describe the spectral nature (pure point or continuous) of a self-similar Sturm-Liouville operator on the line or the half-line. This is motivated by the more general problem of understanding the spectrum of Laplace operators on unbounded finitely ramified self-similar sets. In this context, this furnishes the first example of a description of the spectral nature of the operator in the case where the so-called ``Neumann-Dirichlet'' eigenfunctions are absent. Indiana University Mathematics Journal 2005 text pdf 10.1512/iumj.2005.54.2490 10.1512/iumj.2005.54.2490 en Indiana Univ. Math. J. 54 (2005) 645 - 668 state-of-the-art mathematics http://iumj.org/access/