Extremal properties of eigenvalues for a metric graph

Friedlander, Leonid

doi:10.5802/aif.2095

Cited by 84 publications

(135 citation statements)

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“…This means that the potential is nonnegative and all strengths of vertex interactions are nonnegative.) The proof extends upon a result of L. Friedlander in [6] which gives a lower bounds for Laplace operators on metric graphs with standard vertex conditions. In our notations this case corresponds to I − = I + = 0.…”

supporting

confidence: 50%

“…In this section we follow closely the article [6], where symmetrization technique was applied to obtain estimates for the spectral gap for the standard Laplacian.…”

Section: Symmetrization Techniquementioning

confidence: 99%

“…We first use the same idea as in the first proof and substitute the original Schrödinger operator L on Γ with the Laplace operator L with just two delta interactions of strengths −I − and I + . Estimate for the ground state of L is done following ideas from the article [6].…”

Section: Symmetrization Techniquementioning

confidence: 99%

“…Then an interesting question is the estimate of the spectral gap -the difference between the first two lowest eigenvalues (of course assuming that Γ is connected). In recent papers [6,14] it was proven that among all metric graphs of the same total length, the spectral gap is minimal for the single interval graph. The authors used two different approaches: L. Friedlander applied symmetrization technique, while P. Kurasov and S. Naboko studied behavior of the spectral gap upon topological perturbations of Γ.…”

Section: Introductionmentioning

confidence: 99%

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Schrödinger operators on graphs: Symmetrization and Eulerian cycles

Kurasov

Karreskog

Kupersmidt

2015

Proc. Amer. Math. Soc.

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supporting

confidence: 50%

“…In this section we follow closely the article [6], where symmetrization technique was applied to obtain estimates for the spectral gap for the standard Laplacian.…”

Section: Symmetrization Techniquementioning

confidence: 99%

Section: Symmetrization Techniquementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Schrödinger operators on graphs: Symmetrization and Eulerian cycles

Kurasov

Karreskog

Kupersmidt

2015

Proc. Amer. Math. Soc.

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“…In practice, the function u * can be constructed as the monotone rearrangement of u on the halfline (see [7,17]), and it is clear that…”

Section: Introductionmentioning

confidence: 99%

Ground States for NLS on Graphs: a Subtle Interplay of Metric and Topology

Adami

2016

Math. Model. Nat. Phenom.

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Abstract. We review some recent results on the minimization of the energy associated to the nonlinear Schrödinger Equation on non-compact graphs. Starting from seminal results given by the author together with C. Cacciapuoti, D. Finco, and D. Noja for the star graphs, we illustrate the achiements attained for general graphs and the related methods, developed in collaboration with E. Serra and P. Tilli. We emphasize ideas and examples rather than computations or proofs.

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Operators on Graphs

Borthwick

2020

Graduate Texts in Mathematics

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The spectrum of a finite graph can be defined in terms of its adjacency matrix. In "spectral graph theory," the eigenvalues of this matrix (or related matrices) are used to analyze properties of the graph, such as connectivity.The integer lattice discussed in Section 4.1.4 could be thought of as an infinite discrete graph, with lattice points as vertices and edges connecting nearest neighbors. The discrete Laplacian on Z n , as defined in (4.10), involves only neighboring vertices, and in fact differs from the adjacency matrix by a multiple of the identity. We can thus adapt the formula for Z n to define the discrete Laplacian on an arbitrary graph.Spectral graph theory has a long history as an important tool in combinatorics, with applications to network theory and computer science, as well as number theory, chemistry, and mathematical physics.Another way to build spectral models from graphs is by considering metric graphs, for which a length is assigned to each edge. This identifies the edges with intervals in R, allowing us to consider differential operators acting on a Hilbert space defined as the direct sum of the L 2 spaces for each edge interval. In physical applications, the operator is usually taken to be either the one-dimensional Laplacian or a Schrödinger operator. This combination of a metric graph equipped with a quantum Hamiltonian is called a quantum graph. Research in quantum graphs has been motivated by in large part by physical applications, such as understanding the electromagnetic properties of carbon nanostructures.Note that the key difference between the discrete and continuous cases lies in the support of the functions. In the discrete case the Hilbert space consists of functions on the vertices, while in the quantum graph case they live on the edges. It is possible to synthesize these two approaches and consider functions taking values on both edges and vertices.

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Extremal properties of eigenvalues for a metric graph

Cited by 84 publications

References 1 publication

Schrödinger operators on graphs: Symmetrization and Eulerian cycles

Schrödinger operators on graphs: Symmetrization and Eulerian cycles

Ground States for NLS on Graphs: a Subtle Interplay of Metric and Topology

Operators on Graphs

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