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

Abstract. 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 prob-lem −u∆ ∆ = λuσ, with mixed boundary conditions αu(a) + βu∆(a) = 0 = γu(ρ(b))+ δu∆(ρ(b)). It is known that there exists a sequence of simple eigen-values {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 the Minkowski (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. 1