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.