Spectral and function theory for combinatorial Laplacians

Dodziuk, Józef; Karp, Leon

doi:10.1090/conm/073/954626

Cited by 78 publications

(104 citation statements)

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“…Both boundaries have been considered in the literature, depending on which set plays the role of open set. The first case correspond to consider F as an open set, see for instance [4,9], whereas in the second one this role is played by • F , see for instance [12,15]. The Second Green Identity and the fact that…”

Section: F Is Monotone Which Implies That It Is An Isomorphism and mentioning

confidence: 99%

Potential Theory for Schrödinger operators on finite networks

Bendito¹,

Carmona²,

Encinas³

2005

Rev. Mat. Iberoam.

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We aim here at analyzing the fundamental properties of positive semidefinite Schrödinger operators on networks. We show that such operators correspond to perturbations of the combinatorial Laplacian through 0-order terms that can be totally negative on a proper subset of the network. In addition, we prove that these discrete operators have analogous properties to the ones of elliptic second order operators on Riemannian manifolds, namely the monotonicity, the minimum principle, the variational treatment of Dirichlet problems and the condenser principle. Unlike the continuous case, a discrete Schrödinger operator can be interpreted as an integral operator and therefore a discrete Potential Theory with respect to its associated kernel can be built. We prove that the Schrödinger kernel satisfies enough principles to assure the existence of equilibrium measures for any proper subset. These measures are used to obtain systematic expressions of the Green and Poisson kernels associated with Dirichlet problems.

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Section: F Is Monotone Which Implies That It Is An Isomorphism and mentioning

confidence: 99%

Potential Theory for Schrödinger operators on finite networks

Bendito¹,

Carmona²,

Encinas³

2005

Rev. Mat. Iberoam.

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“…Similarly, one can also show that δ B >c 2 By Lemma 1, we may take a positive solution feL\V) to the equation HJ=λΛHt)f. We have (2.1) <J,(/s), fs} = Σ< Σ /(t(e))β(l(e))u»(e)…”

Section: = --|τί σ F(σt(e))w(e) -M{x) σ λT(β))κ;(β) -(σ W(β))/(σ*)} +mentioning

confidence: 86%

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confidence: 99%

Discrete Schrödinger operators on a graph

Sunada

1992

Nagoya Mathematical Journal

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In this paper, we study some spectral properties of the discrete Schrόdinger operator -Δ + q defined on a locally finite connected graph with an automorphism group whose orbit space is a finite graph.The discrete Laplacian and its generalization have been explored from many different viewpoints (for instance, see [2] [4]). Our paper discusses the discrete analogue of the results on the bottom of the spectrum established by T. Kobayashi, K. Ono and T. Sunada [3] in the Riemannianmanifold-setting. § 1. Discrete Laplacians Let X = (V, E) be a locally finite connected graph without loops and multiple edges. Here V and E are, respectively, the set of vertices and the set of unoriented edges of X. In a natural manner, X is regarded as a one-dimensional CW complex. We assign a positive weight to each vertex and also to each edge by giving mappings m : V->H + and w : E ->R + . Let C 0 (V) and C 0 (E) be the space of all complex-valued functions on V and E with finite support, respectively. Define inner products on CIV) and CIE) byThe completions of C 0 (V) and C Q (E) with respect to those inner products will be denoted by L\V) and D(E), respectively. Each edge has two orientations. We use the symbol E or to represent the set of all oriented edges, so that forgetting orientation yields a twoto-one map p : E oτ -> E. Reversing orientation gives rise to an involution on E or , which we denote by e \->e. We shall use the same symbol w for

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“…Furthermore, it follows from Remark 2.5 that Hθ ,T and H 0,T (= ∆ T ) are unitarily equivalent for any 1-form θ and it is well-known (cf. [3]) that ∆ T d has purely absolutely continuous spectrum, which is given by…”

Section: Yu Higuchi and T Shiraimentioning

confidence: 99%

“…[1], [3], [4], [6]). One of the main topics is to study the relationship between the spectra of discrete Laplacians and the geometries of graphs from many viewpoints.…”

Section: §0 Introductionmentioning

confidence: 99%

Weak Bloch property for discrete magnetic Schrödinger operators

Higuchi

Shirai²

2001

Nagoya Mathematical Journal

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Abstract. For a magnetic Schrödinger operator on a graph, which is a generalization of classical Harper operator, we study some spectral properties: the Bloch property and the behaviour of the bottom of the spectrum with respect to magnetic fields. We also show some examples which have interesting properties. §0. IntroductionThe spectral analysis of a discrete Laplacian, regarded as a discrete analogue of the Laplace-Beltrami operator on a Riemannian manifold, has been investigated by many authors (cf.One of the main topics is to study the relationship between the spectra of discrete Laplacians and the geometries of graphs from many viewpoints. Many researchers have explored the spectra of discrete magnetic Laplacians, each of which is a purturbed Laplacian on a graph by the magnetic field also. For instance, E. Lieb and M. Loss [11] characterized the ground state (in other words, the bottom of the spectrum) of the discrete magnetic Laplacian called a hopping matrix for a finite bipartite planar graph, and T. Sunada [14] gave some criteria for the spectrum which consists of a union of finite number of closed intervals using a twisted group C * -algebra for a discrete magnetic Laplacian with a weak invariance under a group action on an infinite graph.Our main purpose in this paper is to study the spectral properties of the discrete magnetic Laplacian in terms of a growth function for a graph and the behaviour of the bottom of the spectrum as a function of the magnetic flux.

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Spectral and function theory for combinatorial Laplacians

Cited by 78 publications

References 0 publications

Potential Theory for Schrödinger operators on finite networks

Potential Theory for Schrödinger operators on finite networks

Discrete Schrödinger operators on a graph

Weak Bloch property for discrete magnetic Schrödinger operators

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