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 xγ 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.
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