Spectral estimates for infinite quantum graphs

Abstract

We investigate the bottom of the spectra of infinite quantum graphs, i.e., Laplace operators on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then prove the Cheeger-type estimate. Our definition of the isoperimetric constant is purely combinatorial and thus it establishes connections with the combinatorial isoperimetric constant, one of the central objects in spectral graph theory and in the theory of simple random walks on graphs. The latter enables us to prove a number of criteria for quantum graphs to be uniformly positive or to have purely discrete spectrum. We demonstrate our findings by considering trees, antitrees and Cayley graphs of finitely generated groups.2010 Mathematics Subject Classification. Primary 34B45; Secondary 35P15; 81Q35.

Symmetry breaking in two-dimensional square grids: persistence and failure of the dimensional crossover

“…However, we reported it here explicitly both because it is of actual help for the proofs of some of our next theorems, and because it provides a first condition which is equivalent to the validity of the two-dimensional Sobolev inequality. We also mention that isoperimetric problems on grids and more general planar graphs have been widely considered in the discrete setting (see for instance [9,10,20,21,41,62]), and investigations on related issues started recently also in the metric framework [42,49].…”

“…Nowadays, a major line of research concerns noncompact metric graphs with infinitely many bounded edges and, in particular, periodic graphs, driven by their applications to the study of new materials such as for instance carbon nanutubes and graphene (see, e.g., [14,45]). In the linear framework, the spectral analysis of Schrödinger operators on periodic structures is a focal topic both in the metric (see [8,17,33,42] and references therein) and in the discrete setting [37], also in connection with the wider subject of periodic elliptic operators (see [44] for an overview on this point).…”

Self‐adjoint and Markovian extensions of infinite quantum graphs

“…If $${\mathcal {G}}_d$$ is the edge graph of a tessellation of $${\mathbb {R}}^2$$, then positivity of $$\alpha ({\mathcal {G}})$$ can be deduced from certain curvature‐type quantities [65]. On the other hand, by [55, Corollary 4.5(i)], $$\lambda _0({\mathcal {G}})>0$$ holds true if the combinatorial isoperimetric constant of $${\mathcal {G}}_d$$ is positive and $$\ell ^\ast ({\mathcal {G}}) := \sup _{e\in {\mathcal {E}}}|e|<\infty$$. For example, this holds true if $${\mathcal {G}}_d$$ is an infinite tree without leaves [55, Lemma 8.1] or if $${\mathcal {G}}_d$$ is a Cayley graph of a non‐amenable finitely generated group [55, Lemma 8.12(i)].…”

“…Note that every function $$f \in {\mathbb {H}}({\mathcal {G}})$$ is uniquely determined by its vertex values $${{\bf f}}:= f|_{\mathcal {V}}=(f(v))_{v \in {\mathcal {V}}}$$. Recall also the following result (see, for example, [55, Equation (2.32)]). Lemma Let $${\mathcal {G}}$$ be a metric graph satisfying the assumptions in Hypothesis 2.1.…”

The Krein–von Neumann Extension for Schrödinger Operators on Metric Graphs

“…A typical application of surgery principles for graph eigenvalues consists of deriving spectral inequalities in terms of geometric and topological parameters of the graph, such as its total length, diameter, number of edges or vertices, or its first Betti number (or Euler characteristics, equivalently). For a few recent advances on spectral inequalities for quantum graphs, we refer to [10,14,28,31,39,42]. To demonstrate how surgery principles for the perturbed Krein Laplacian on a metric graph may be applied, we establish lower bounds for the positive eigenvalues, in terms of eigenvalues of a loop graph or edge lengths.…”