On a spectral analysis for the Sierpinski gasket

Fukushima, Masatoshi; Shima, T.

doi:10.1007/bf00249784

Cited by 270 publications

(275 citation statements)

References 8 publications

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“…Description of µ = µ N D in the discrete case It is useful to make the change of variable v = [34], and in subsequent works, [19], [49].…”

Section: The Sierpinski Gasketmentioning

confidence: 99%

“…In particular, he computed explicitly the eigenvalues and showed the existence of the so-called molecular states (that we call Neumann-Dirichlet eigenfunctions in this text) which are eigenfunctions with compact support. This was made rigorous and generalized to the continuous operator defined on the Sierpinski gasket itself by Fukushima and Shima (cf [19]). The spectral type of the operator on the Sierpinski lattice, has been analyzed by Teplyaev, cf [49].…”

Section: Introductionmentioning

confidence: 99%

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Spectral properties of self-similar lattices and iteration of rational maps

Sabot¹

2003

Mémoires Soc. Math. France

121

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Abstract:In this text we consider discrete Laplace operators defined on lattices based on finitely-ramified self-similar sets, and their continuous analogous defined on the self-similar sets themselves. We are interested in the spectral properties of these operators. The basic example is the lattice based on the Sierpinski gasket. We introduce a new renormalization map which appears to be a rational map defined on a smooth projective variety (more precisely, this variety is isomorphic to a product of three types of Grassmannians: complex Grassmannians, Lagrangian Grassmannians, orthogonal Grassmannians). We relate some characteristics of the dynamics of its iterates with some characteristics of the spectrum of our operator. More specifically, we give an explicit formula for the density of states in terms of the Green current of the map, and we relate the indeterminacy points of the map with the so-called Neumann-Dirichlet eigenvalues which lead to eigenfunctions with compact support on the unbounded lattice. Depending on the asymptotic degree of the map we can prove drastically different spectral properties of the operators. Our formalism is valid for the general class of finitely ramified self-similar sets (i.e. for the class of pcf selfsimilar sets of Kigami, cf [25]). Hence, this work aims at a generalization and a better understanding of the initial work of the physicists Rammal and Toulouse on the Sierpinski gasket (cf [35], [34]).Résumé: Dans ce texte, nous considérons le Laplacien discret, défini sur un réseau(1) E-mail address: sabot@ccr.jussieu.fr 1 2 construità partir d'un ensemble auto-similaire finiment ramifié, et son analogue continu défini sur l'ensemble auto-similaire lui-même. Nous nous intéressons aux propriétés spectrales de ces opérateurs. L'exemple le plus classique est celui du triangle de Sierpinski (Sierpinski gasket) et du réseau discret associé. Nous introduisons une nouvelle application de renormalisation qui se trouveêtre une application rationnelle définie sur une variété projective lisse (plus précisément, cette variété est un produit de Grassmaniennes de trois types: Grassmaniennes classiques, Grassmaniennes Lagrangiennes, Grassmaniennes orthogonales). Nous relions certaines propriétés spectrales de ces opérateurs avec la dynamique des itérés de cette application. En particulier, nous donnons une formule explicite de la densité d'états en termes du courant de Green de l'application, et nous caractérisons le spectre de Neumann-Dirichlet (qui correspond aux fonctions propresà support compact sur l'ensemble infini)à l'aide des points d'indétermination de l'application. Suivant le degré asymptotique de l'application nous pouvons prouver que les propriétés spectrales de l'opérateur sont très différentes. Notre formalisme s'appliqueà la classe des ensembles auto-similaires finiment ramifiés (ou autrement dità la classe des "pcf self-similar sets" de Kigami, cf [25]

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“…Description of µ = µ N D in the discrete case It is useful to make the change of variable v = [34], and in subsequent works, [19], [49].…”

Section: The Sierpinski Gasketmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Spectral properties of self-similar lattices and iteration of rational maps

Sabot¹

2003

Mémoires Soc. Math. France

121

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“…Using the spectral decimation method of Rammal and Toulouse [11], Fukushima and Shima gave a complete characterization of the spectrum of the Laplacian on SG [5]. Further description of this spectrum has been investigated in [4,15].…”

Section: Introduction and Motivationsmentioning

confidence: 99%

Asymptotics of eigenvalue clusters for Schrödinger operators on the Sierpiński gasket

Okoudjou

Strichartz

2007

Proc. Amer. Math. Soc.

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Abstract. In this note we investigate the asymptotic behavior of spectra of Schrödinger operators with continuous potential on the Sierpiński gasket SG. In particular, using the existence of localized eigenfunctions for the Laplacian on SG we show that the eigenvalues of the Schrödinger operator break into clusters around certain eigenvalues of the Laplacian. Moreover, we prove that the characteristic measure of these clusters converges to a measure. Results similar to ours were first observed by A. Weinstein and V. Guillemin for Schrödinger operators on compact Riemannian manifolds.

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“…Similarly, we can show that if no, then choose a new independent set by adding the cell that is not in the span and, recursively, we can construct a linearly independent set of cell measures. As we shall see, we need only go to level two for SG 4 . SG 4 is the fractal K satisfying We need to determine the harmonic extension matrices that extend the values on V 0 of a harmonic function to the vertices of V 1 (as displayed in Figure 13).…”

Section: Then This Is Equal Tomentioning

confidence: 99%

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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On a spectral analysis for the Sierpinski gasket

Cited by 270 publications

References 8 publications

Spectral properties of self-similar lattices and iteration of rational maps

Spectral properties of self-similar lattices and iteration of rational maps

Asymptotics of eigenvalue clusters for Schrödinger operators on the Sierpiński gasket

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

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