Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
| ||

``Probabilités et Modèles Aléatoires''
| ||

**Auteur(s): **

**Code(s) de Classification MSC:**

- 82B44 Disordered systems (random Ising models, random Schrodinger operators, etc.)
- 60H25 Random operators and equations, See also {47B80}
- 28A80 Fractals, See also {58Fxx}

**Résumé:** Starting from a finitely ramified self-similar set $X$
we can construct an unbounded set $X_{<\infty>}$
by blowing-up the initial set $X$.
We consider random blow-ups and prove elementary
properties of the spectrum of the
natural Laplace operator on $X_{<\infty>}$ (and on the associated lattice).
We prove that the spectral type of the operator is almost surely
deterministic with the blow-up and that the spectrum coincides
with the support of the density of states almost surely
(actually our result is more precise).
We also prove that if the density of states is completely created by
the so-called Neuman-Dirichlet eigenvalues, then almost surely
the spectrum is
pure point with compactly supported eigenfunctions.

**Mots Clés:** *Spectral theory of Schrödinger operators ; random self-adjoint operators ;
analysis on self-similar sets*

**Date:** 2001-06-18

**Prépublication numéro:** *PMA-671*

**Postscript file : **PMA-671.ps