Eigenvalue distribution of second-order dynamic equations on time scales considered as fractals

Let T ⊂ [a, b] be a time scale with a, b ∈ T. In this paper we study the asymptotic distribution of eigenvalues of the following linear problem −u = λuσ , with mixed boundary conditions αu(a) + βu (a) = 0 = γ u(ρ(b)) + δu (ρ(b)). It is known that there exists a sequence of simple eigenvalues {λk } k; we consider the spectral counting function N(λ,T) = #{k: λk λ}, and we seek for its asymptotic expansion as a power of λ. Let d be theMinkowski (or box) dimension of T, which gives the order of growth of the number K(T, ε) of intervals of length ε needed to cover T, namely K(T, ε) ≈ εd . We prove an upper bound of N(λ) which involves the Minkowski dimension, N(λ,T) Cλd/2, where C is a positive constant depending only on the Minkowski content of T (roughly speaking, its d-volume, although the Minkowski content is not a measure). We also consider certain limiting cases (d = 0, infinite Minkowski content), and we show a family of self similar fractal sets where N(λ,T) admits two-side estimates. © 2008 Elsevier Inc. All rights reserved.