## DEPENDENCE OF THE SPECTRUM OF A QUANTUM GRAPH ON VERTEX CONDITIONS AND EDGE LENGTHS

### BibTeX

@MISC{Berkolaiko_dependenceof,

author = {Gregory Berkolaiko and Peter Kuchment},

title = {DEPENDENCE OF THE SPECTRUM OF A QUANTUM GRAPH ON VERTEX CONDITIONS AND EDGE LENGTHS},

year = {}

}

### OpenURL

### Abstract

Abstract. We study the dependence of the quantum graph Hamiltonian, its resolvent, and its spectrum on the vertex conditions and graph edge lengths. In particular, several results on the interlacing (bracketing) of the spectra of graphs with different vertex conditions are obtained and their applications are discussed. 1.

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Citation Context ...nd study their interlacing properties. Eigenvalue interlacing (or bracketing) is a powerful tool in spectral theory with such well-known applications as the derivation of the asymptotic Weyl law, see =-=[5]-=-. In the graph setting, it allows one to estimate eigenvalue of a given graph via the eigenvalues of its subgraphs, which may be easier to calculate. Interlacing results on graphs have already been us... |

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Citation Context .... Vertex conditions. We will briefly describe now the known descriptions of the vertex conditions one can add to the differential expression (2) in order to create a self-adjoint operator (see, e.g., =-=[8, 10, 12, 14]-=- for details). Assume that the domain of the operator is a subspace of the Sobolev space ˜ ... |

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Citation Context ...with the boundedness of each of the spectra from below). This proves the statement of the theorem. □ 6.2. Nodal count on trees. Here we present a simple proof of a result that was first discovered in =-=[1, 18, 21]-=-. With all ground-setting work done in Section 5 the proof is significantly shortened. Theorem 6.4. Let ... |

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Citation Context ...t allows one to estimate eigenvalue of a given graph via the eigenvalues of its subgraphs, which may be easier to calculate. Interlacing results on graphs have already been used in several situations =-=[2, 17, 21]-=-. We significantly generalize these results and put them in the form particularly suited for the applications. We discuss several applications. In particular, we give a simple proof of the number of n... |

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Citation Context ...t allows one to estimate eigenvalue of a given graph via the eigenvalues of its subgraphs, which may be easier to calculate. Interlacing results on graphs have already been used in several situations =-=[2, 17, 21]-=-. We significantly generalize these results and put them in the form particularly suited for the applications. We discuss several applications. In particular, we give a simple proof of the number of n... |

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Citation Context ...ge is irrelevant. This is not so true anymore if one wants to include derivative term of an odd order, e.g. magnetic potential, but we shall not address such operators in the present note (see, e.g., =-=[8, 19]-=- concerning these issues). The natural smoothness requirement coming from the ODE theory is that ... |

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Citation Context ... thin wires, eigenvalue statistics of chaotic systems and properties of the nodal domains of eigenfunctions. We refer the interested reader to the recent reviews [9, 13, 15] and collections of papers =-=[3, 7]-=-. A quantum graph is a metric graph equipped with a self-adjoint differential “Hamiltonian” operator (usually of Schrödinger type) defined on the edges and matching conditions specified at the vertice... |

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