Localization for the Anderson Model on Trees with Finite Dimensions

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We study the Laplacian on family preserving metric graphs. These are graphs that have a certain symmetry that, as we show, allows for a decomposition into a direct sum of one-dimensional operators whose properties are explicitly related to the structure of the graph. Such decompositions have been extremely useful in the study of Schrödinger operators on metric trees. We show that the tree structure is not essential, and moreover, obtain a direct and simple correspondence between such decompositions in the discrete and continuum case.operators whose structure is directly related to the structure of the tree and to the boundary conditions at the vertices. A similar method exists in the discrete case, where the graph is considered as a combinatorial object and the operator studied is the discrete Laplacian or the adjacency matrix (see, e.g., [3,9,10,12]). While the similarity between the decomposition in the continuous and discrete case is clear and lies in the exploitation of the symmetry properties of the graph, it is important to note there are essential technical differences. Whereas the discrete case involves studying cyclic subspaces generated by specially chosen functions, the continuum case (as presented in [28,32]) involves defining the relevant invariant subspaces directly and relies heavily on the structure of the tree.It has recently been realized in [13] that in the discrete case, the tree structure is not essential for this decomposition. It is in fact possible to carry out this procedure for a more general class of graphs that we call 'family preserving' and whose definition we give below (see Definition 2.2) 1 . This class contains radially symmetric trees, antitrees (see Section 5 for the definition), and various other graphs.We refer the reader to [13] for examples and some graphics.The objective of this work is to extend this decomposition for family preserving graphs to the continuum case of metric graphs as well. Since, as remarked above, the standard decomposition technique for metric trees relies heavily on the tree structure, one needs to take a different approach. A natural approach to this task, and the one that we shall take, is to try and obtain a direct translation of the discrete decomposition to the decomposition in the metric case. Such an approach has also the added bonus of making explicit the connection between the combinatorial and continuum decompositions, and as we shall show, can recover the original Naimark-Solomyak procedure from the procedure in the discrete case.While employing the discrete decomposition scheme is indeed natural for getting a decomposition in the metric case, the results in this paper should by no means be viewed as a direct extension of those in [7]. First, as the functions in the metric case live on edges, whereas those in the discrete case live on vertices, it is a non-trivial task to adapt one procedure to the other. Second, as is evident in Section 4 (where we describe the decomposition algorithm) the discrete decomposition is only one of several steps...

Section: Introductionmentioning

confidence: 99%

On the Decomposition of the Laplacian on Metric Graphs

Breuer

Levi

2020

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Absolutely Continuous Spectrum for Random Schrödinger Operators on Tree-Strips of Finite Cone Type

Sadel

2012

Anderson Transition at Two-Dimensional Growth Rate on Antitrees and Spectral Theory for Operators with One Propagating Channel

Sadel

2015

We show that the Anderson model has a transition from localization to delocalization at exactly 2 dimensional growth rate on antitrees with normalized edge weights which are certain discrete graphs. The kinetic part has a one-dimensional structure allowing a description through transfer matrices which involve some Schur complement. For such operators we introduce the notion of having one propagating channel and extend theorems from the theory of one-dimensional Jacobi operators that relate the behavior of transfer matrices with the spectrum. These theorems are then applied to the considered model. In essence, in a certain energy region the kinetic part averages the random potentials along shells and the transfer matrices behave similar as for a one-dimensional operator with random potential of decaying variance. At d dimensional growth for d > 2 this effective decay is strong enough to obtain absolutely continuous spectrum, whereas for some uniform d dimensional growth with d < 2 one has pure point spectrum in this energy region. At exactly uniform 2 dimensional growth also some singular continuous spectrum appears, at least at small disorder. As a corollary we also obtain a change from singular spectrum (d ≤ 2) to absolutely continuous spectrum (d ≥ 3) for random operators of the type Pr∆ d Pr + λV on Z d , where Pr is an orthogonal radial projection, ∆ d the discrete adjacency operator (Laplacian) on Z d and λV a random potential. *