Asymptotics of eigenvalue clusters   for Schrödinger operators  on the Sierpiński gasket

Okoudjou, Kasso A.; Strichartz, Robert S.

doi:10.1090/s0002-9939-07-09008-9

Cited by 10 publications

(5 citation statements)

References 15 publications

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“…We also present graphical data from numerical approximations of lower portions of the spectra. Similar results for the spectrum of Schrödinger operators −∆ + V are given in [14].…”

Section: E(u)supporting

confidence: 77%

Conformal energy, conformal Laplacian, and energy measures on the Sierpinski gasket

Azzam

Hall

Strichartz

2007

Trans. Amer. Math. Soc.

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Abstract. On the Sierpinski Gasket (SG) and related fractals, we define a notion of conformal energy E ϕ and conformal Laplacian ∆ ϕ for a given conformal factor ϕ, based on the corresponding notions in Riemannian geometry in dimension n = 2. We derive a differential equation that describes the dependence of the effective resistances of E ϕ on ϕ. We show that the spectrum of ∆ ϕ (Dirichlet or Neumann) has similar asymptotics compared to the spectrum of the standard Laplacian, and also has similar spectral gaps (provided the function ϕ does not vary too much). We illustrate these results with numerical approximations. We give a linear extension algorithm to compute the energy measures of harmonic functions (with respect to the standard energy), and as an application we show how to compute the L p dimensions of these measures for integer values of p ≥ 2. We derive analogous linear extension algorithms for energy measures on related fractals.

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“…We also present graphical data from numerical approximations of lower portions of the spectra. Similar results for the spectrum of Schrödinger operators −∆ + V are given in [14].…”

Section: E(u)supporting

confidence: 77%

Conformal energy, conformal Laplacian, and energy measures on the Sierpinski gasket

Azzam

Hall

Strichartz

2007

Trans. Amer. Math. Soc.

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“…Our work is part of a long term study of mathematical physics on fractals and graphs, more specifically, quantum Hall systems with AMO and their topological quantum phases [1][2][3][4][5][6]8,28,35,[48][49][50], in which novel features of physical systems can be associated with the unusual spectral and geometric properties of fractals and graphs compared to smooth manifolds.…”

Section: Introductionmentioning

confidence: 99%

Spectral decimation of piecewise centrosymmetric Jacobi operators on graphs

Mograby¹,

Balu²,

Okoudjou³

et al. 2022

Preprint

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We study the spectral theory of a class of piecewise centrosymmetric Jacobi operators defined on an associated family of substitution graphs. Given a finite centrosymmetric matrix viewed as a weight matrix on a finite directed path graph and a probabilistic Laplacian viewed as a weight matrix on a locally finite strongly connected graph, we construct a new graph and a new operator by edge substitution. Our main result proves that the spectral theory of the piecewise centrosymmetric Jacobi operator can be explicitly related to the spectral theory of the probabilistic Laplacian using certain orthogonal polynomials. Our main tools involve the so-called spectral decimation, known from the analysis on fractals, and the classical Schur complement. We include several examples of self-similar Jacobi matrices that fit into our framework.

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“…Remark 2. Observe that in comparison with (14), the error in ( 19) is decaying at a slower rate. This is a consequence of the fact that, at the optimal N, the eigenfunctions that are not localized at scale N make up a larger proportion (in terms of dimension) of the space E Λ than they do in the spaces E λ with λ ≈ Λ.…”

Section: General Szegö Theorem On Sgmentioning

confidence: 99%

“…For a fixed large j we may then choose N such that the bound in (17) is minimized, which occurs when ǫ ≈ 3 N −j . Setting (3/5) N α = 3 N −j we compute 3 N −j = 3 −jα log(5/3) log 3+α log (5/3) and substitute into (17) to obtain (14), using d j ≈ 3 j . Remark 1.…”

Section: Szegö Limit Theorem On Sg For a Single Eigenspacementioning

confidence: 99%