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On The Spectrum Of Infinite Quantum Graphs

Marco Düfel, James B. Kennedy, Delio Mugnolo, Marvin Plümer, Matthias Täufer

Abstract

We study the interplay between spectrum, geometry and boundary conditions for two distinguished self-adjoint realisations of the Laplacian on infinite metric graphs, the so-called riedrichs and Neumann extensions. We introduce a new criterion for compactness of the resolvent and apply this to identify a transition from purely discrete to non-empty essential spectrum among a class of infinite metric graphs, a phenomenon that seems to have no known counterpart for Laplacians on Euclidean domains of infinite volume. In the case of discrete spectrum we then prove upper and lower bounds on eigenvalues, thus extending a number of bounds previously only known in the compact setting to infinite graphs. Some of our bounds, for instance in terms of the inradius, are new even on compact graphs.

On The Spectrum Of Infinite Quantum Graphs

Abstract

We study the interplay between spectrum, geometry and boundary conditions for two distinguished self-adjoint realisations of the Laplacian on infinite metric graphs, the so-called riedrichs and Neumann extensions. We introduce a new criterion for compactness of the resolvent and apply this to identify a transition from purely discrete to non-empty essential spectrum among a class of infinite metric graphs, a phenomenon that seems to have no known counterpart for Laplacians on Euclidean domains of infinite volume. In the case of discrete spectrum we then prove upper and lower bounds on eigenvalues, thus extending a number of bounds previously only known in the compact setting to infinite graphs. Some of our bounds, for instance in terms of the inradius, are new even on compact graphs.
Paper Structure (17 sections, 33 theorems, 135 equations, 6 figures)

This paper contains 17 sections, 33 theorems, 135 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Infinite metric and quantum graphs
  3. Notation and elementary properties
  4. Compact exhaustions
  5. Lebesgue and Sobolev spaces
  6. Laplace-type operators on infinite graphs
  7. Embedding theorems and discreteness of the spectrum
  8. Embedding theorems for Sobolev spaces
  9. Spectral phase transition on graphs of infinite volume
  10. Spectral approximation
  11. Lower bounds on the lowest positive eigenvalue
  12. Symmetrisation techniques for isoperimetric inequalities
  13. Further lower bounds
  14. Diameter
  15. Inradius
  16. ...and 2 more sections

Key Result

Lemma 2.9

Let $\mathcal{G}$ be a locally finite, connected metric graph, let $\mathsf{v} \in \mathsf{V}$ be any vertex of $\mathcal{G}$, let $(\mathcal{G}_{\mathsf{v},n})_{n\in\mathbb{N}}$ be the compact exhaustion from Example ExampleApprox, and let $(\mathcal{G}_n)_{n\in\mathbb{N}}$ be any other compact exh Moreover, $k_2 \to \infty$, and $k_1$ can be chosen to tend to $\infty$, as $n \to \infty$.

Figures (6)

  • Figure 2.1: The infinite ladder graph.
  • Figure 2.2: From left to right, the graphs $\mathcal{G}_1$, $\mathcal{G}_2$ and $\mathcal{G}_3$.
  • Figure 3.1: The diagonal comb graph $\mathcal{G}_{\frac{1}{2}}$ and its two centre vertices, see Definition \ref{['def:centre_vertex']} and Remark \ref{['rem:two:centres']}.
  • Figure 4.1: The four cases where equality in \ref{['eq:bkkm1']} holds. Dirichlet conditions (at vertices or ends) are coloured light grey, standard/Neumann conditions are depicted as filled black circles. (1) In the top graph the Friedrichs realisation leads to a formal Dirichlet condition at the end on the right side of the infinite necklace; (2) the second graph is a compact necklace with a Dirichlet condition at one end and a standard condition at the other; (3) in the third row there is a Dirichlet vertex of degree two at the finite end of the necklace and a Neumann condition at the other end; (4) the bottom graph has a Dirichlet condition at the end on the left and a Neumann condition at the end on the right.
  • Figure 4.2: A partition of a graph $\mathcal{G}$ into two subgraphs $\mathcal{G}_1$ and $\mathcal{G}_2$. To apply Lemma \ref{['lem:cutting_Dirichlet']} the derivatives of the eigenfunction $\psi$ at the vertices $\mathsf{v}_1$ and $\mathsf{v}_2$ pointing into the respective edges $\mathsf{e}_1$, $\mathsf{e}_2$ and $\mathsf{e}_3$ have to be nonpositive.
  • ...and 1 more figures

Theorems & Definitions (90)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4: Topological end
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Example 2.8
  • Lemma 2.9
  • Proposition 2.10
  • ...and 80 more