Essential Self-adjointness for Combinatorial Schrödinger Operators II-Metrically non Complete Graphs

Verdìère, Yves Colin de; Torki-Hamza, Nabila; Truc, Francoise

doi:10.1007/s11040-010-9086-7

Cited by 59 publications

(67 citation statements)

References 18 publications

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“…It is straightforward to check that the corresponding path pseudo metric ̺ p is intrinsic (see [55, Example 2.1], [59]). (b) Another pseudo metric was suggested in [25]. Namely, let ̺ be a path pseudo metric generated by the function p : E → (0, ∞)…”

Section: Quantum Graphs With Kirchhoff Vertex Conditionsmentioning

confidence: 99%

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Spectral Theory of Infinite Quantum Graphs

Exner

Kostenko

Malamud

et al. 2018

Ann. Henri Poincaré

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We investigate quantum graphs with infinitely many vertices and edges without the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths of its edges. Our central result is a close connection between spectral properties of a quantum graph and the corresponding properties of a certain weighted discrete Laplacian on the underlying discrete graph. Using this connection together with spectral theory of (unbounded) discrete Laplacians on infinite graphs, we prove a number of new results on spectral properties of quantum graphs. Namely, we prove several self-adjointness results including a Gaffney type theorem. We investigate the problem of lower semiboundedness, prove several spectral estimates (bounds for the bottom of spectra and essential spectra of quantum graphs, CLR-type estimates) and study spectral types. Contents2010 Mathematics Subject Classification. Primary 81Q35; Secondary 34B45; 34L05.

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Section: Quantum Graphs With Kirchhoff Vertex Conditionsmentioning

confidence: 99%

“…is finite, is uniformly Lipschitz with respect to the metric ̺ 1/2 and hence admits a continuation F to V as a Lipschitz function. Following [25], we set f ∞ := F ↾ V ∞ . On the other hand, the latter is further equivalent to the fact that (V; ̺ m ) is not complete as a metric space.…”

Section: Quantum Graphs With Kirchhoff Vertex Conditionsmentioning

confidence: 99%

Spectral Theory of Infinite Quantum Graphs

Exner

Kostenko

Malamud

et al. 2018

Ann. Henri Poincaré

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“…The essential self-adjointness of L q (G) on weighted locally finite graphs have been studied in [6,18]. We are interested in the study of the Schrödinger eigenvalue problem with Dirichlet boundary condition with respect to finite vertex set S, namely,…”

Section: (24)mentioning

confidence: 99%

Maximization of combinatorial Schrödinger operator’s smallest eigenvalue with Dirichlet boundary condition

Mohanty

Lal

2015

Discrete Mathematics

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a b s t r a c tFor a nonnegative potential function q and a given locally finite graph G, we study the combinatorial Schrödinger operator L q (G) = ∆ G + q with Dirichlet boundary condition on a proper finite subset S of the vertex set of G such that the induced subgraph on S is con-We prove the existence and uniqueness of the maximizer of the smallest Dirichlet eigenvalue of L q (G), whenever the potential function q ∈ Υ p . Furthermore, we also establish the analogue of the Euler-Lagrange equation on graphs.

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“…In the case q ≡ 1, the weighted distance (8) was defined in [4]. We say that the metric space (G, d w,a;q ) is complete if every Cauchy sequence of vertices has a limit in V .…”

Section: A Magnetic Schrödinger Operatormentioning

confidence: 99%

“…This result of [3] was extended in [11] to semi-bounded below operator H| C c (V ) in ℓ 2 w (V ), where H is as in (6). For a study of essential self-adjointness of Schrödinger operators (without magnetic field) on a metrically non-complete graph, see [4]. For a generalization of [4, Theorem 3.1], see [21,Corollary 3.6].…”

Section: Background Of the Problemmentioning

confidence: 99%

A Sears-type self-adjointness result for discrete magnetic Schrödinger operators

Milatovic

2012

Journal of Mathematical Analysis and Applications

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a b s t r a c tIn the context of a weighted graph with vertex set V and bounded vertex degree, we give a sufficient condition for the essential self-adjointness of the operator ∆ σ + W , where ∆ σ is the magnetic Laplacian and W : V → R is a function satisfying W (x) ≥ −q(x) for all x ∈ V , with q: V → [1, ∞). The condition is expressed in terms of completeness of a metric that depends on q and the weights of the graph. The main result is a discrete analogue of the results of I. Oleinik and M.A. Shubin in the setting of non-compact Riemannian manifolds.

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Essential Self-adjointness for Combinatorial Schrödinger Operators II-Metrically non Complete Graphs

Abstract: We consider weighted graphs, we equip them with a metric structure given by a weighted distance, and we discuss essential self-adjointness for weighted graph Laplacians and Schrödinger operators in the metrically non complete case.

Cited by 59 publications

References 18 publications

Spectral Theory of Infinite Quantum Graphs

Spectral Theory of Infinite Quantum Graphs

Maximization of combinatorial Schrödinger operator’s smallest eigenvalue with Dirichlet boundary condition

A Sears-type self-adjointness result for discrete magnetic Schrödinger operators

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