• DOI:10.1007/s11005-021-01404-2
• Corpus ID: 234681557

Spectral Hausdorff dimensions for a class of Schrödinger operators in bounded intervals

@inproceedings{Bazo2021SpectralHD,
  title={Spectral Hausdorff dimensions for a class of Schr{\"o}dinger operators in bounded intervals},
  author={Vanderl{\'e}a Rodrigues Baz{\~a}o and T O Carvalho and C{\'e}sar R. de Oliveira},
  year={2021}
}

• V. R. Bazão, T. O. Carvalho, C. R. de Oliveira
• Published 13 May 2021
• Mathematics

Exact Hausdorff dimensions are computed for singular continuous components of the spectral measures of a class of Schrödinger operators in bounded intervals. 1. Main results We are interested in Hausdorff dimensional properties of spectral measures of Schrödinger operators (1.1) (Hu)(x) = − u dx2 (x) + V (x)u(x) acting in L(Ib), where Ib = [0, b], 0 < b <∞, is a bounded interval of R; our potentials V (x) are signed combs of delta distributions carefully spaced in Ib and accumulating only at b. The boundary condition at 0 is (1.2) u(0) cos(φ) + u(0) sin(φ) = 0, with φ ∈ [0, 2π) fixed. In [14], Pearson has presented a family of such Schrödinger operators −d2/dx2 + V0(x), on the bounded interval Ib, which has purely absolutely continuous spectrum in a certain range of energies (the potential V0(x) is a selected comb of delta distributions and its construction is recalled in Appendix A). Borrowing ideas from [12, 15], we will perturb such potential V0(x) to obtain a model (1.1) with a singular continuous spectral component in this interval (see also Example 14.6.j in [16]). The main contribution of this work is to compute the Hausdorff dimensions of their spectral measures. It is usually hard to present examples of potentials for which one can say something about fractal properties of spectral measures. To the best of our knowledge, these are the first examples, in bounded intervals, of singular continuous Schrödinger operators with computable dimensions. We need some preparation in order to state our main results. 1.1. Hausdorff dimension of measures. If μ is a positive and finite Borel measure on R, its lower local dimension at x ∈ support(μ) is given by d−μ (x) = lim inf ǫ↓0 lnμ((x− ǫ, x+ ǫ)) ln ǫ . 1 2 VANDERLÉA R. BAZAO, TÚLIO O. CARVALHO, AND CÉSAR R. DE OLIVEIRA The general idea is to estimate the scaling property μ((x− ǫ, x+ ǫ)) ∼ ǫμ for small ǫ > 0. Let dimH(S) denote the Hausdorff dimension of the set S ⊂ R and 0 < α ≤ 1; the upper Hausdorff dimension of μ is defined as dim+H(μ) = inf{dimH(S);μ(R \ S) = 0, S a Borel subset of R}, and its lower Hausdorff dimension as dim−H(μ) = sup{α;μ(S) = 0 if dimH(S) < α, S a Borel subset of R}. If A ⊂ R is a Borel set, we shall also consider the dimensions of the restriction of μ to such set, that is, μ;A(·) := μ(A ∩ ·). It turns out that [5, 13] (1.3) dim−H(μ) = μ-ess.inf d − μ , dim + H(μ) = μ-ess.sup d − μ . Hence, the information of the lower local dimensions of μ gives the values of its Hausdorff dimensions. 1.2. The potential. Let the potential V0(x) be a comb of delta distributions at suitable points an ∈ (0, b), as constructed in [14, 15, 16], for which the Schrödinger operator (1.1) has purely absolutely continuous spectrum on an interval J ⊂ R. We consider the potential (1.4) Vω(x) = V0(x) + ∞