Related papers: Spectra of "fattened" open book structures
Lattice structures play a central role in spectral graph theory, offering analytical insight into diffusion, synchronization, and transport processes on regular discrete spaces. While their spectral properties are completely characterized…
We study the spectral stability of a 2D discrete Schr\"{o}dinger equation on a square lattice, in the simultaneous presence of a fractional Laplacian and $\cal{PT}$ symmetry. For that purpose, we compute the plane-wave spectrum in closed…
We consider a metric graph $\mathcal{G}$ made of two graphs $\mathcal{G}_1$ and $\mathcal{G}_2$ attached at one point. We derive a formula relating the spectral determinant of the Laplace operator $S_\mathcal{G}(\gamma)=\det(\gamma-\Delta)$…
Nonlinear spectral graph theory is an extension of the traditional (linear) spectral graph theory and studies relationships between spectral properties of nonlinear operators defined on a graph and topological properties of the graph…
Quantum graphs have become in this century a favorite playground for mathematicians, mathematical physicists, and chemists, due to their manifold applications as models of thin structures, as well as presenting sometimes simpler playground…
We describe the spectral theory of the adjacency operator of a graph which is isomorphic to homogeneous trees at infinity. Using some combinatorics, we reduce the problem to a scattering problem for a finite rank perturbation of the…
Quantum confinement endows two-dimensional (2D) layered materials with exceptional physics and novel properties compared to their bulk counterparts. Although certain two- and few-layer configurations of graphene have been realized and…
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 and natural Laplacians acting on it and…
Quantum graphs have attracted attention from mathematicians for some time. A quantum graph is defined by having a Laplacian on each edge of a metric graph and imposing boundary conditions at the vertices to get an eigenvalue problem. A…
We investigate the limit properties of a family of Schr\"odinger operators of the form $H_\varepsilon= -\frac{\mathrm{d}^2}{\mathrm{d}x^2}+ \frac{\lambda(\varepsilon)}{\varepsilon^2}Q \big(\frac{x}{\varepsilon}\big)$ acting on $n$-edge star…
multiplication operator on a Hilbert space may be approximated with finite sections by choosing an orthonormal basis of the Hilbert space. Nonzero multiplication operators on $L^2$ spaces of functions are never compact and then such…
Determining and analyzing the spectra of graphs is an important and exciting research topic in theoretical computer science. The eigenvalues of the normalized Laplacian of a graph provide information on its structural properties and also on…
Using the theory of extensors developed in a previous paper we present a theory of the parallelism structure on arbitrary smooth manifold. Two kinds of Cartan connection operators are introduced and both appear in intrinsic versions (i.e.,…
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 venerable history as they are related to…
In this article, we continue the discussion of Fang-Wu (2015) to estimate the spectral gap of the Ornstein-Uhlenbeck operator on path space over a Riemannian manifold of pinched Ricci curvature. Along with explicit estimates we study the…
For a compact, connected metric graphs with a boundary that consists of $k$ vertices, we prove that an arbitrary symmetric $k\times k$ matrix with real entries can be realized as the Dirichlet-to-Neumann operator for the Laplacian plus a…
The spectral theory of the Laplace differential operator for biregular quantum graphs is developed. Trees are studied in detail. Generating functions for closed non backtracking walks appear when resolvents for trees are related to…
We derive a necessary condition for compactness of the weighted $\overline\partial$-Neumann operator on the space $L^2(\mathbb C^n,e^{-\varphi})$, under the assumption that the corresponding weighted Bergman space of entire functions has…
We study some spectral properties of a matrix that is constructed as a combination of a Laplacian and an adjacency matrix of simple graphs. The matrix considered depends on a positive parameter, as such we consider the implications in…
Branched n-coverings of Riemann surfaces are described by a 2d lattice gauge theory of the symmetric group S(n) defined on a cell discretization of the surface. We study the theory in the large-n limit, and we find a rich phase diagram with…