  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.
