Refinement of the spectral asymptotics of generalized Krein Feller operators

Uta Freiberg

doi:10.1515/form.2011.017

Article

Refinement of the spectral asymptotics of generalized Krein Feller operators

Uta Freiberg

Published/Copyright: April 2, 2011

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From the journal Volume 23 Issue 2

Abstract

Spectral asymptotics of operators of the form are investigated. In the case of self-similar measures μ and ν it turns out that the eigenvalue counting function N(x) under both Dirichlet and Neumann conditions behaves like x^γ as x → ∞, where the spectral exponent γ is given in terms of the scaling numbers of the measures. More precisely, it holds that

In the present paper, we give a refinement of this spectral result, i.e. we give a sufficient condition under which the term N(x)x^–γ converges. We show, using renewal theory, that the behaviour of N(x)x^–γ depends essentially on whether the set of logarithms of the scaling numbers of the measures is arithmetic.

Keywords.: Krein Feller operator; Sturm Liouville operator; measure geometric Laplacian; spectral asymptotics; integrated density of states; renewal theory; self-similarity; Cantor like sets

Received: 2009-01-29

Published Online: 2011-04-02

Published in Print: 2011-March

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Articles in the same Issue

https://doi.org/10.1515/form.2011.017

Keywords for this article

Krein Feller operator; Sturm Liouville operator; measure geometric Laplacian; spectral asymptotics; integrated density of states; renewal theory; self-similarity; Cantor like sets