Spectral estimates for infinite quantum graphs

@article{Kostenko2018SpectralEF,
  title={Spectral estimates for infinite quantum graphs},
  author={Aleksey Kostenko and Noema Nicolussi},
  journal={Calculus of Variations and Partial Differential Equations},
  year={2018},
  volume={58},
  pages={1-40}
}
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… 
Self‐adjoint and Markovian extensions of infinite quantum graphs
We investigate the relationship between one of the classical notions of boundaries for infinite graphs, graph ends, and self‐adjoint extensions of the minimal Kirchhoff Laplacian on a metric graph.
Lower estimates on eigenvalues of quantum graphs
A method for estimating the spectral gap along with higher eigenvalues of nonequilateral quantum graphs has been introduced by Amini and Cohen-Steiner recently: it is based on a new transference
Quantum graphs on radially symmetric antitrees
We investigate spectral properties of Kirchhoff Laplacians on radially symmetric antitrees. This class of metric graphs enjoys a rich group of symmetries, which enables us to obtain a decomposition
Strong Isoperimetric Inequality for Tessellating Quantum Graphs
We investigate isoperimetric constants of infinite tessellating metric graphs. We introduce a curvature-like quantity, which plays the role of a metric graph analogue of discrete curvature notions
Spectral Monotonicity for Schrödinger Operators on Metric Graphs
We study the influence of certain geometric perturbations on the spectra of self-adjoint Schr\"odinger operators on compact metric graphs. Results are obtained for radially symmetric vertex
A theory of spectral partitions of metric graphs
TLDR
An abstract framework for the study of clustering in metric graphs is introduced: after suitably metrising the space of graph partitions, Laplacians are restricted to the clusters thus arising and use their spectral gaps to define several notions of partition energies.
Upper eigenvalue bounds for the Kirchhoff Laplacian on embedded metric graphs
We derive upper bounds for the eigenvalues of the Kirchhoff Laplacian on a compact metric graph depending on the graph's genus g. These bounds can be further improved if $g = 0$, i.e. if the metric
Upper Eigenvalue Bounds for the Kirchhoff Laplacian on Embbeded Metric Graphs
We derive upper bounds for the eigenvalues of the Kirchhoff Laplacian on a compact metric graph depending on the graph’s genus g. These bounds can be further improved if g = 0, i.e. if the metric
Laplacians on infinite graphs: discrete vs continuous
There are two main notions of a Laplacian operator associated with graphs: discrete graph Laplacians and continuous Laplacians on metric graphs (widely known as quantum graphs). Both objects have a
...
1
2
...

References

SHOWING 1-10 OF 71 REFERENCES
Spectral Theory of Infinite Quantum Graphs
We investigate quantum graphs with infinitely many vertices and edges without the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths
On the Spectral Gap of a Quantum Graph
We consider the problem of finding universal bounds of “isoperimetric” or “isodiametric” type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices,
Infinite quantum graphs
Infinite quantum graphs with δ-interactions at vertices are studied without any assumptions on the lengths of edges of the underlying metric graphs. A connection between spectral properties of a
Edge connectivity and the spectral gap of combinatorial and quantum graphs
We derive a number of upper and lower bounds for the first nontrivial eigenvalue of Laplacians on combinatorial and quantum graph in terms of the edge connectivity, i.e. the minimal number of edges
Eigenvalue estimates on quantum graphs
On a finite connected metric graph, we establish upper bounds for the eigenvalues of the Laplacian. These bounds depend on the length, the Betti number, and the number of pendant vertices. For trees,
Quantum Graphs which Optimize the Spectral Gap
A finite discrete graph is turned into a quantum (metric) graph once a finite length is assigned to each edge and the one-dimensional Laplacian is taken to be the operator. We study the dependence of
Spectral analysis of metric graphs and related spaces
The aim of the present article is to give an overview of spectral theory on metric graphs guided by spectral geometry on discrete graphs and manifolds. We present the basic concept of metric graphs
Some relations between analytic and geometric properties of infinite graphs
Volume growth, spectrum and stochastic completeness of infinite graphs
We study the connections between volume growth, spectral properties and stochastic completeness of locally finite weighted graphs. For a class of graphs with a very weak spherical symmetry we give a
...
1
2
3
4
5
...