Eigenvalue bracketing for discrete and metric graphs

Lledó, Fernando; Post, Olaf

doi:10.1016/j.jmaa.2008.07.029

Cited by 32 publications

(43 citation statements)

References 23 publications

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“…. More precisely, the multiplicity of (πk) 2 is related to the global topology of the graph, as explained in [48]. Related multiplicity calculations for quantum graphs were carried out in [49].…”

Section: Fig 5 the Kagome Lattice And A Finitely Supported Eigenfunmentioning

confidence: 99%

Unique continuation principles and their absence for Schrödinger eigenfunctions on combinatorial and quantum graphs and in continuum space

Peyerimhoff¹,

Täufer²,

Veselić³

2017

Nanosystems: Phys. Chem. Math.

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For the analysis of the Schrödinger and related equations it is of central importance whether a unique continuation principle (UCP) holds or not. In continuum Euclidean space, quantitative forms of unique continuation imply Wegner estimates and regularity properties of the integrated density of states (IDS) of Schrödinger operators with random potentials. For discrete Schrödinger equations on the lattice, only a weak analog of the UCP holds, but it is sufficient to guarantee the continuity of the IDS. For other combinatorial graphs, this is no longer true. Similarly, for quantum graphs the UCP does not hold in general and consequently, the IDS does not need to be continuous.

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Section: Fig 5 the Kagome Lattice And A Finitely Supported Eigenfunmentioning

confidence: 99%

Unique continuation principles and their absence for Schrödinger eigenfunctions on combinatorial and quantum graphs and in continuum space

Peyerimhoff¹,

Täufer²,

Veselić³

2017

Nanosystems: Phys. Chem. Math.

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“…We start this section with a short excursion into cohomology (for the standard vertex space and its oriented version, see also [25]). Assume that d = d G : G −→ 2 (E) is an exterior derivative for the vertex space G .…”

Section: Indices For Dirac Operators On Discrete Graphsmentioning

confidence: 99%

“…In [35] we also discussed the "enlarged" case. Finally, in [25], the "exceptional" Dirichlet eigenvalues arising from eigenfunctions on an equilateral metric graph vanishing at the vertices are described in a geometric way.…”

Section: O Postmentioning

confidence: 99%

“…Results on spectral theory of discrete or combinatorial Laplacians can be found, e.g., in [41,4,5,7,27]. For continuous (quantum) graph Laplacians we mention the works [12,15,[17][18][19][20][21]25,29,31,34,36,38] and the recent book [8]. In particular, a heat equation approach for the index formula for certain metric graph Laplacian (with energy-independent scattering matrix) can be found in [16].…”

Section: Introductionmentioning

confidence: 99%

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First Order Approach and Index Theorems for Discrete and Metric Graphs

Post

2009

Ann. Henri Poincaré

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The aim of the present paper is to introduce a unified notion of Laplacians on discrete and metric graphs. In order to cover all self-adjoint vertex conditions for the associated metric graph Laplacian, we develop systematically a new type of discrete graph operators acting on a decorated graph. The decoration at each vertex of degree d is given by a subspace of C d , generalising the fact that a function on the standard vertex space has only a scalar value. We illustrate the abstract concept by giving classical examples throughout the article. Our approach includes infinite graphs as well. We develop the notion of exterior derivative, differential forms, Dirac and Laplace operators in the discrete and metric case, using a supersymmetric framework. We calculate the (supersymmetric) index of the discrete Dirac operator generalising the standard index formula involving the Euler characteristic of a graph. Finally, we show that for finite graphs, the corresponding index for the metric Dirac operator agrees with the discrete one.

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“…Therefore it is often possible to compare different form domains while the domains of the representing operators remain unrelated. This fact allows, e.g., to develop spectral bracketing techniques in very different mathematical and physical situations using the language of quadratic forms [33,34].…”

Section: Introductionmentioning

confidence: 99%

Self-adjoint extensions of the Laplace–Beltrami operator and unitaries at the boundary

Ibort

Lledó

Pérez-Pardo

2015

Journal of Functional Analysis

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We construct in this article a class of closed semi-bounded quadratic forms on the space of square integrable functions over a smooth Riemannian manifold with smooth compact boundary. Each of these quadratic forms specifies a semibounded self-adjoint extension of the Laplace-Beltrami operator. These quadratic forms are based on the Lagrange boundary form on the manifold and a family of domains parametrized by a suitable class of unitary operators on the boundary that will be called admissible. The corresponding quadratic forms are semi-bounded below and closable. Finally, the representing operators correspond to semi-bounded self-adjoint extensions of the Laplace-Beltrami operator. This family of extensions is compared with results existing in ✩ 635 Boundary conditions the literature and various examples and applications are discussed.

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Eigenvalue bracketing for discrete and metric graphs

Cited by 32 publications

References 23 publications

Unique continuation principles and their absence for Schrödinger eigenfunctions on combinatorial and quantum graphs and in continuum space

Unique continuation principles and their absence for Schrödinger eigenfunctions on combinatorial and quantum graphs and in continuum space

First Order Approach and Index Theorems for Discrete and Metric Graphs

Self-adjoint extensions of the Laplace–Beltrami operator and unitaries at the boundary

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