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Pure Point Spectrum for the Laplacian on Unbounded Nested Fractals

Authors
Journal
Journal of Functional Analysis
0022-1236
Publisher
Elsevier
Publication Date
Volume
173
Issue
2
Identifiers
DOI: 10.1006/jfan.2000.3567
Keywords
  • Spectral Theory
  • Schrödinger Operators
  • Diffusions On Fractals

Abstract

Abstract We prove for the class of nested fractals introduced by T. Lindstrøm (1990, Memoirs Amer. Math. Soc. 420) that the integrated density of states is completely created by the so-called Neuman–Dirichlet eigenvalues. The corresponding eigenfunctions lead to eigenfunctions with compact support on the unbounded set and we prove that for a large class of blow-ups the set of Neuman–Dirichlet eigenfunctions is complete, leading to a pure point spectrum with compactly supported eigenfunctions. This generalizes previous results of H. Teplyaev (1998, J. Funct. Anal. 159, 537–567) on the Sierpinski Gasket. Our methods are elementary and use only symmetry arguments via the representations of the symmetry group of the set.

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