Spectral Asymptotics of Generalized Measure Geometric Laplacians on Cantor Like Sets

Freiberg, Uta

doi:10.1515/form.2005.17.1.87

Cited by 47 publications

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“…This is a key result in order to get estimates on the asymptotics of eigenvalues as in the Weyl's classical theorem for the Laplacian on bounded Euclidean domains (see [42], [43], [10], [36][Chapter XIII .15]). We point out that in [17], [19], [24] [40] the authors study the asymptotics of the eigenvalues for the Laplacian defined on self-similar geometric objects. Lapidus (cf.…”

Section: Introductionmentioning

confidence: 99%

Spectral analysis of 1D nearest-neighbor random walks and applications to subdiffusive trap and barrier models

Faggionato

2012

Electron. J. Probab.

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We consider a sequence X-(n), n >= 1, of continuous-time nearest-neighbor random walks on the one dimensional lattice Z. We reduce the spectral analysis of the Markov generator of X((n)) with Dirichlet conditions outside (0, n) to the analogous problem for a suitable generalized second order differential operator - D-mn D-x, with Dirichlet conditions outside a given interval. If the measures dm(n) weakly converge to some measure dm(infinity), we prove a limit theorem for the eigenvalues and eigenfunctions of - D-mn D-x to the corresponding spectral quantities of - D-m infinity D-x. As second result, we prove the Dirichlet-Neumann bracketing for the operators - D-m D-x and, as a consequence, we establish lower and upper bounds for the asymptotic annealed eigenvalue counting functions in the case that m is a self-similar stochastic process. Finally, we apply the above results to investigate the spectral structure of some classes of subdiffusive random trap and barrier models coming from one-dimensional physics

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Section: Introductionmentioning

confidence: 99%

Spectral analysis of 1D nearest-neighbor random walks and applications to subdiffusive trap and barrier models

Faggionato

2012

Electron. J. Probab.

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“…We impose The operators μ and the associated eigenvalue problems have been studied in connection with spectral functions of strings and diffusion processes (see [8,9,3,32,22] and the references therein). More recently, they have also been studied in connection with fractal measures μ (see [5,[13][14][15][16][17]20,33]). We also note that if μ is absolutely continuous, (1.3) reduces to a well-known Sturm-Liouville equation.…”

Section: Introductionmentioning

confidence: 99%

Eigenvalues and eigenfunctions of one-dimensional fractal Laplacians defined by iterated function systems with overlaps

Chen

Ngai

2010

Journal of Mathematical Analysis and Applications

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Keywords:Eigenvalues Eigenfunctions Fractal Laplacian Self-similar measure Iterated function system with overlaps Second-order self-similar identities Finite element method Under the assumption that a self-similar measure defined by a one-dimensional iterated function system with overlaps satisfies a family of second-order self-similar identities introduced by Strichartz et al., we obtain a method to discretize the equation defining the eigenvalues and eigenfunctions of the corresponding fractal Laplacian. This allows us to obtain numerical solutions by using the finite element method. We also prove that the numerical eigenvalues and eigenfunctions converge to the true ones, and obtain estimates for the rates of convergence. We apply this scheme to the fractal Laplacians defined by the well-known infinite Bernoulli convolution associated with the golden ratio and the 3-fold convolution of the Cantor measure. The iterated function systems defining these measures do not satisfy the open set condition or the post-critically finite condition; we use secondorder self-similar identities to analyze the measures.

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“…Those operators are introduced in [4]; their spectral asymptotics are obtained in [5]. Operators Af ≡ A m f and related Markov processes, where supp µ is not necessarily the whole interval [0, 1], have been investigated by many authors, see, for example, [2], [3], [6], [9], [10], [11], [12].…”

Section: Introductionmentioning

confidence: 99%

Zeros of eigenfunctions of a class of generalized second order differential operators on the Cantor set

Freiberg

Löbus

2004

Mathematische Nachrichten

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Spectral Asymptotics of Generalized Measure Geometric Laplacians on Cantor Like Sets

Cited by 47 publications

References 0 publications

Spectral analysis of 1D nearest-neighbor random walks and applications to subdiffusive trap and barrier models

Spectral analysis of 1D nearest-neighbor random walks and applications to subdiffusive trap and barrier models

Eigenvalues and eigenfunctions of one-dimensional fractal Laplacians defined by iterated function systems with overlaps

Zeros of eigenfunctions of a class of generalized second order differential operators on the Cantor set

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