Quantum Graphs which Optimize the Spectral Gap

Band, Ram; Lévy, Guillaume

doi:10.1007/s00023-017-0601-2

Cited by 49 publications

(63 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, m(v) = deg(v) for all v ∈ V if the corresponding metric graph is equilateral (i.e., |e| ≡ 1), and hence (6.3) coincides with the definition suggested for combinatorial Laplacians in [21]. Notice that for equilateral graphs (6.1) reads 4) and hence in this case 2…”

Section: Moreover For Any Non-boundary Pointmentioning

confidence: 70%

Spectral estimates for infinite quantum graphs

Kostenko

Nicolussi

2018

Calc. Var.

View full text Add to dashboard Cite

We investigate the bottom of the spectra of infinite quantum graphs, i.e., Laplace operators on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then prove the Cheeger-type estimate. Our definition of the isoperimetric constant is purely combinatorial and thus it establishes connections with the combinatorial isoperimetric constant, one of the central objects in spectral graph theory and in the theory of simple random walks on graphs. The latter enables us to prove a number of criteria for quantum graphs to be uniformly positive or to have purely discrete spectrum. We demonstrate our findings by considering trees, antitrees and Cayley graphs of finitely generated groups.2010 Mathematics Subject Classification. Primary 34B45; Secondary 35P15; 81Q35.

show abstract

Section: Moreover For Any Non-boundary Pointmentioning

confidence: 70%

Spectral estimates for infinite quantum graphs

Kostenko

Nicolussi

2018

Calc. Var.

View full text Add to dashboard Cite

show abstract

“…For a general compact metric graph of total length L, this eigenvalue was shown by Nicaise [Nic87] to be no smaller than π 2 /L 2 , with equality if and only if G is a path (i.e., interval); see [Sol02,Fri05a,KN14] for further proofs. Recently, Band and Lévy [BaLe17] obtained a stronger lower bound under the assumption that the graph is doubly connected: the non-trivial eigenvalue is no lower than 4π 2 /L 2 ; see also [BKKM17] for a sharper estimate in the case of higher connectivities.…”

mentioning

confidence: 99%

Surgery principles for the spectral analysis of quantum graphs

Berkolaiko

Kennedy

Kurasov

et al. 2019

Trans. Amer. Math. Soc.

116

View full text Add to dashboard Cite

We present a systematic collection of spectral surgery principles for the Laplacian on a compact metric graph with any of the usual vertex conditions (natural, Dirichlet or δ-type), which show how various types of changes of a local or localised nature to a graph impact on the spectrum of the Laplacian. Many of these principles are entirely new; these include "transplantation" of volume within a graph based on the behaviour of its eigenfunctions, as well as "unfolding" of local cycles and pendants. In other cases we establish sharp generalisations, extensions and refinements of known eigenvalue inequalities resulting from graph modification, such as vertex gluing, adjustment of vertex conditions and introducing new pendant subgraphs.To illustrate our techniques we derive a new eigenvalue estimate which uses the size of the doubly connected part of a compact metric graph to estimate the lowest nontrivial eigenvalue of the Laplacian with natural vertex conditions. This quantitative isoperimetric-type inequality interpolates between two known estimates -one assuming the entire graph is doubly connected and the other making no connectivity assumption (and producing a weaker bound) -and includes them as special cases. Contents2010 Mathematics Subject Classification. 34B45 (05C50 35P15 81Q35).

show abstract

“…[7] and references therein), such continuum models (also known as quantum graphs) have drawn the attention of the spectral theory community and have served as a platform for the study of various topics. These include trace formulas in quantum chaos [19], isospectrality and its association with geometry [21], Anderson localization and extended states [1,2,11,22,24,33], Hardy inequalities [16,28], eigenvalue estimates [5,8,17] and others [14,15,26]. A useful method in the context of infinite metric trees (i.e.…”

Section: Introductionmentioning

confidence: 99%

On the Decomposition of the Laplacian on Metric Graphs

Breuer

Levi

2020

Ann. Henri Poincaré

View full text Add to dashboard Cite

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...

show abstract

Quantum Graphs which Optimize the Spectral Gap

Cited by 49 publications

References 28 publications

Spectral estimates for infinite quantum graphs

Spectral estimates for infinite quantum graphs

Surgery principles for the spectral analysis of quantum graphs

On the Decomposition of the Laplacian on Metric Graphs

Contact Info

Product

Resources

About