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

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Schrödinger operators on fractal lattices with random blow-ups


Code(s) de Classification MSC:

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