## Eigenvalue bracketing for discrete and metric graphs

Venue: | J. Math. Anal. Appl |

Citations: | 4 - 2 self |

### BibTeX

@ARTICLE{Lledó_eigenvaluebracketing,

author = {Fernando Lledó and Olaf Post},

title = {Eigenvalue bracketing for discrete and metric graphs},

journal = {J. Math. Anal. Appl},

year = {},

pages = {806--833}

}

### OpenURL

### Abstract

Abstract. We develop eigenvalue estimates for the Laplacians on discrete and metric graphs using different types of boundary conditions at the vertices of the metric graph. Via an explicit correspondence of the equilateral metric and discrete graph spectrum (also in the “exceptional” values of the metric graph corresponding to the Dirichlet spectrum) we carry over these estimates from the metric graph Laplacian to the discrete case. We apply the results to covering graphs and present examples where the covering graph Laplacians have spectral gaps. 1.

### Citations

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Citation Context ...Dirichlet Laplacian ˇ ∆∂V (G,m). Similarly, if m(V ) is infinite, then the implication (i) ⇒ (ii) is still valid. Proof. The proof of the equivalence for ∂V = ∅ and finite graphs can be found e.g. in =-=[Ch97]-=-; the case ∂V ̸= ∅ follows similarly. If G has finite mass, then the constant function V is in ℓ 2 (V, m), and the argument for finite graphs carries over. If m(V ) is infinite, then the spectral symm... |

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Citation Context ... ′ (1) = 0 only enter in the operator domain by requiring the boundary terms to vanish which appear after partial integration. For details, we refer to [RS80, Sec. VIII.6] and [RS78, Sec. XIII.15] or =-=[D95]-=-. Any other (linear) boundary condition, like e.g. the ϑ-equivariant condition f(1) = e iϑ f(0) leads to a space domh ϑ between dom h D and dom h N (the action of h ϑ being the same, namely h ϑ (f) = ... |

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Citation Context ...let and Neumann eigenvalues of a suitable chosen fundamental domain were close to each other (cf. [LP08, Thm. 3.3]). An upper bound is given once the covering group has positive Kadison constant (see =-=[Sun92]-=-). Homology groups have also been used for metric graph Laplacians with magnetic field, see [KS03] for details. The type of spectrum for magnetic Laplacians on a metric equilateral square lattice was ... |

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Citation Context ...er Laplacians with Dirichlet conditions on a subset ∂V of the vertices. In this case, the relative homology group H 1 (X, ∂V ) enters. The multiplicities of the eigenvalues were already calculated in =-=[vB85]-=- by a direct proof without using the homology groups. The advantage of using homology groups is that is can be generalised to other types of vertex boundary conditions (like Dirichlet and equivariant)... |

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Citation Context ...LP08, Thm. 3.3]). An upper bound is given once the covering group has positive Kadison constant (see [Sun92]). Homology groups have also been used for metric graph Laplacians with magnetic field, see =-=[KS03]-=- for details. The type of spectrum for magnetic Laplacians on a metric equilateral square lattice was analysed in [BGP07], and, in particular, for irrational flux, the spectrum has Cantor structure. M... |

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Citation Context ...ers to a metric graph together with a self-adjoint differential operator as a quantum graph. Recent interesting reviews on discrete geometric analysis and quantum graphs can be found in [Sun08] resp. =-=[Kuc08]-=- (see also references therein). The aim of the present paper is to use spectral results for the metric graph to obtain spectral information of the discrete Laplacian. In particular, we will obtain res... |

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Citation Context ...ondition giving the relation with the discrete graph (or at least with its homology), as we have already noticed for the vertex-based eigenfunctions in Proposition 4.1. Remark 5.3. Note that Cattaneo =-=[Ca97]-=- already calculated the spectrum of an equilateral (possibly infinite) graph (with ∂V = ∅) also for the exceptional values Σ D without taking care about the multiplicities. She obtains the same result... |

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Citation Context ...groups have also been used for metric graph Laplacians with magnetic field, see [KS03] for details. The type of spectrum for magnetic Laplacians on a metric equilateral square lattice was analysed in =-=[BGP07]-=-, and, in particular, for irrational flux, the spectrum has Cantor structure. Magnetic Laplacians may be seen as a generalisation of equivariant Laplacians for Abelian coverings treated in detail in S... |

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Citation Context ...etails). For Abelian groups, the Floquet-Bloch decomposition can be used in order to calculate the spectrum of the operator on the covering, leading to a detailed analysis in certain models, see e.g. =-=[KP07]-=- for hexagonal lattices (modeling carbon nano-structures). We refer to the intervals Ik = Ik(Y, ∂V ) as Kirchhoff-Dirichlet (KD) intervals. Note that they depend usually on the fundamental domain. The... |

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Citation Context ...on is of the form L 2 (X) ∼ = ∫ ⊕ bΓ L 2 (Y ), ∆ X ∼ = ∫ ⊕ bΓ ∆ ρ X0 . Since Γ is Abelian, ρ can be parametrised by ϑ ∈ Rr via ρ(γ) = eiϑ·γ . We also write λϑ k details we refer to [Sun08, Sec. 6] or =-=[LP07]-=- and the references therein. Moreover, from the direct integral decomposition and the continuous dependence of λ ρ k on ρ, we deduce for the spectrum of the Kirchhoff Laplacian σ(∆X) = ⋃ ρ∈ b Γ is cal... |

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Citation Context ...ght. We will need the operators ¯ d and ¯ d ∗ in Section 5. For more details and a general concept, in which the oriented and unoriented version of an exterior derivative embed naturally, we refer to =-=[P07b]-=- (see also [P07a, P07c]). As for the oriented exterior derivative, we can also define a Dirichlet version of ¯ d, namely, e∈Ev ¯d0: ℓ ∂V 2 (V, m) −→ ℓ 2 (E, m), ¯ d0 := ¯ d ◦ ι. As before, we have ¯ d... |

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Citation Context ...ion of H. Note that if G0 is bipartite then any fundamental domain H is, but not vice versa. Proof. The first statement is analogue to the one of Proposition 4.1 and can be shown similarly as e.g. in =-=[P07a]-=-. The proof of the second statement is similar to the proofs of Lemma 4.3 and Proposition 4.7. We only sketch the ideas here. Let f ∈ N ρ (λn) be an eigenfunction, interpreted as function on a fundame... |

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Citation Context ..., the spectral relation of Proposition 4.1 is still true, even more, one can show that all spectral types (discrete and essential, absolutely and singular continuous, (pure) point) are preserved, see =-=[BGP08]-=- for details. Moreover, N ∂V (λn) = N ∂V 0 (λn) □16 FERNANDO LLEDÓ AND OLAF POST (n ≥ 1) due to the fact that the trivial vertex based eigenfunctions ϕn are no longer in L 2 (X). Moreover, we can eas... |

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(Show Context)
Citation Context ...d antiperiodic equivariant eigenvalues (ϑ = 0 and ϑ = π) give already the band edges. For groups with more than one generator, the band edges need not to be on the boundary of the Brillouin zone, see =-=[HKSW07]-=- and appear as KD eigenvalues, but with alternating role (Bk = [λ0 k , λπk ] for k = 1, 3 and B2 = [λπ 2 , λ2, 0]). This phenomena also appears for Schrödinger operators (see [KP07] and the references... |

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