Schrödinger operators on fractal lattices with random blow-ups

Details Schrödinger operators on fractal lattices with random blow-ups Schrödinger operators on fractal lattices with random blow-ups

2002-01-18

arxiv.org/abs/math-ph/0201041v1

Physics

Mathematical Physics

15 pages

Scientific paper

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.

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