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 problem −u = λu σ , 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 {λ k } k ; we consider the spectral counting function N(λ, T) = #{k: λ k 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) ≈ ε d . We prove an upper bound of N(λ) which involves the Minkowski dimension, N(λ, T) Cλ d/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.

Detailed asymptotic of eigenvalues on time scales

“…[3]. Indeed, we show in Remark (2.4) below that our proof here gives very bad estimates for the sets with positive dimension considered in our previous work.…”

“…0, and our argument in this work cannot replace the one in Ref. [3]. Let us recall that the set T ¼ {n 2a } n < {0} has Minkowski dimension equal to (a þ 1) 21 , and the upper bound obtained previously was Nðl; TÞ # Cl 1=2ðaþ1Þ :…”

“…. }), have the same Minkowski dimension ln(2)/ln (3). Let us also observe that for any d [ (0,1), there exists a sequence {a n } n converging to zero with Minkowski dimension equal to d, since the set…”

“…In our previous work [3], we have studied the eigenvalue counting function N(l, T) defined as Nðl; TÞ ¼ #{k : l k # l};…”

“…In Ref. [3] we replaced it by b, considering the eigenvalue problem forb ¼ r 2 ðbÞ. Note that the boundary condition uðrðbÞÞ ¼ 0 implies that the number of (generalized) zeros between b and s 2 (b) is lower than 3, so replacing q(t) by l 5 , it is seen that…”