Localization on Quantum Graphs with Random Vertex Couplings

Klopp, Frédéric; Pankrashkin, Konstantin

doi:10.1007/s10955-008-9517-z

Cited by 12 publications

(15 citation statements)

References 36 publications

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“…Indeed, for this model, if all but one random variables are fixed, the perturbation is of rank one. Using [17], the study of the spectrum of this model is reduce to the study of a family of energy-dependent discrete operator with random potential. Let L ∈ N and (ω n ) be non-negative random variables with a common compactly supported bounded density.…”

Section: One-dimensional Quantum Graphs With Random Vertex Couplingmentioning

confidence: 99%

Decorrelation Estimates for Random Discrete Schrödinger Operators in Dimension One and Applications to Spectral Statistics

Shirley

2014

J Stat Phys

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Section: One-dimensional Quantum Graphs With Random Vertex Couplingmentioning

confidence: 99%

Decorrelation Estimates for Random Discrete Schrödinger Operators in Dimension One and Applications to Spectral Statistics

Shirley

2014

J Stat Phys

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“…We will prove both estimates, theorems 4 and 5 by exploiting a correspondence between the quantum graphs and discrete operators. A similar approach was used in [25] for quantum graphs with random coupling constants and more details on the reduction can be found there. (v)).…”

Section: Multiscale Analysis and Finite-volume Operatorsmentioning

confidence: 99%

“…that the bottom of the spectrum is pure point with exponentially decaying eigenfunctions.There are two basic methods of proving localization for random operators: the multiscale analysis going back to Fröhlich and Spencer [16] and the Aizenman-Molchanov method [3]. The Aizenman-Molchanov method gives explicit and efficient criteria for localization in terms of the Green function but only works under special assumptions on the way the randomness enters the problem (we used this method for the study of the random coupling model in [25]), which does not hold in the situation we are studying. On the other hand, the multiscale analysis is a rather universal tool which can handle very abstract situations [38].…”

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confidence: 99%

Localization on Quantum Graphs with Random Edge Lengths

Klopp

Pankrashkin

2009

Lett Math Phys

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The spectral properties of the Laplacian on a class of quantum graphs with random metric structure are studied. Namely, we consider quantum graphs spanned by the simple Z d -lattice with δ-type boundary conditions at the vertices, and we assume that the edge lengths are randomly independently identically distributed. Under the assumption that the coupling constant at the vertices does not vanish, we show that the operator exhibits the Anderson localization near the spectral edges situated outside a certain forbidden set. MSC 2000: 81Q10; 35R60; 47B80; 60H25Up to now, the most studied models of disordered media are either discrete (tight-binding) Hamiltonians or continuous Schrödinger operators. In the last decade, there is an increasing interest in the analysis of quantum Hamiltonians on so-called quantum graphs, i.e. differential operators on singular one-dimensional spaces, see the collection of papers [6,13,15,27]. A quantum graph is composed of one-dimensional differential operators on the edges and boundary conditions at the vertices describing coupling of edges. Such operators provide an effective model for the study of various phenomena in the condensed matter physics admitting an experimental verification, and there is natural question about the influence of random perturbations in such systems [39]. There are numerous possibilities to introduce randomness: combinatorial structure, coefficients of differential expression, coupling, metric, etc. Being locally of one-dimensional nature and admitting a complex global shape, quantum graphs take an intermediate position between the one-dimensional and higher dimensional Schrödinger operators. It seems that the paper [26] considering the random necklace model was the first one discussing random interactions and the Anderson localization in the quantum graph setting. Later, these results were generalized for radial tree configurations [22], where Anderson localization at all energies was proved. Both papers used a machinery specific for one-dimensional operators. The paper [1] addressed the spectral analysis on quantum tree graphs with random edge lengths; it appears that the Anderson localization does not hold near the bottom of the spectrum at least in the weak disorder limit and one always has some absolutely continuous spectrum. Another important class of quantum graphs is given by Z d -lattices. The paper [14] studied the situation where each edge carries a random potential and showed the Anderson localization near the bottom of the spectrum. Some generalizations were then obtained in [20,21]. The case of random coupling was considered by the present authors in [25]; recently we learned on an earlier paper [10] where some preliminary estimates for the same model were obtained.The present Letter is devoted to the study of quantum graphs spanned by the Z d -lattice where the edge lengths are random independent identically distributed variables. We consider the free Laplacian on each edge with δ-type boundary conditions and show, under certain technical assump...

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“…We refer e.g. to [18][19][20][21][22][23][24][25][26][27][28][29][30][31] for generalizations to more general differential operators and for the analysis of particular configurations. The aim of the present paper is to improve the relation (2).…”

Section: Introductionmentioning

confidence: 99%

Unitary dimension reduction for a class of self-adjoint extensions with applications to graph-like structures

Pankrashkin

2012

Journal of Mathematical Analysis and Applications

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a b s t r a c tWe consider a class of self-adjoint extensions using the boundary triplet technique. Assuming that the associated Weyl function has the special form M(z) = (m(z)Id − T )n(z) −1 with a bounded self-adjoint operator T and scalar functions m, n we show that there exists a class of boundary conditions such that the spectral problem for the associated self-adjoint extensions in gaps of a certain reference operator admits a unitary reduction to the spectral problem for T . As a motivating example we consider differential operators on equilateral metric graphs, and we describe a class of boundary conditions that admit a unitary reduction to generalized discrete Laplacians.

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Localization on Quantum Graphs with Random Vertex Couplings

Cited by 12 publications

References 36 publications

Decorrelation Estimates for Random Discrete Schrödinger Operators in Dimension One and Applications to Spectral Statistics

Decorrelation Estimates for Random Discrete Schrödinger Operators in Dimension One and Applications to Spectral Statistics

Localization on Quantum Graphs with Random Edge Lengths

Unitary dimension reduction for a class of self-adjoint extensions with applications to graph-like structures

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