Relating Zeta Functions of Discrete and Quantum Graphs
| Autor: | J. M. Harrison, Tracy Weyand |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
05C99
81Q10 81Q35 Dirichlet conditions 010102 general mathematics FOS: Physical sciences Statistical and Nonlinear Physics Mathematical Physics (math-ph) Mathematics::Spectral Theory Equilateral triangle 01 natural sciences Complete bipartite graph Riemann zeta function Vertex (geometry) Combinatorics symbols.namesake Quantum graph 0103 physical sciences symbols 0101 mathematics 010306 general physics Laplace operator Eigenvalues and eigenvectors Mathematical Physics Mathematics |
| DOI: | 10.48550/arxiv.1612.04273 |
| Popis: | We write the spectral zeta function of the Laplace operator on an equilateral metric graph in terms of the spectral zeta function of the normalized Laplace operator on the corresponding discrete graph. To do this, we apply a relation between the spectrum of the Laplacian on a discrete graph and that of the Laplacian on an equilateral metric graph. As a by-product, we determine how the multiplicity of eigenvalues of the quantum graph, that are also in the spectrum of the graph with Dirichlet conditions at the vertices, depends on the graph geometry. Finally we apply the result to calculate the vacuum energy and spectral determinant of a complete bipartite graph and compare our results with those for a star graph, a graph in which all vertices are connected to a central vertex by a single edge. Comment: 12 pages; added reference after Corollary 1 (on page 4) |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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