Related papers: Quantum graphs with the Bethe-Sommerfeld property
The paper is concerned with the number of open gaps in spectra of periodic quantum graphs. The well-known conjecture by Bethe and Sommerfeld (1933) says that the number of open spectral gaps for a system periodic in more than one direction…
We locate gaps in the spectrum of a Hamiltonian on a periodic cuboidal (and generally hyperrectangular) lattice graph with $\delta$ couplings in the vertices. We formulate sufficient conditions under which the number of gaps is finite. As…
We investigate spectral properties of quantum graphs in the form of a periodic chain of rings with a connecting link between each adjacent pair, assuming that wave functions at the vertices are matched through conditions manifestly…
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 of its edges. Our central result is a…
Let $\Gamma$ be an arbitrary $\mathbb{Z}^n$-periodic metric graph, which does not coincide with a line. We consider the Hamiltonian $\mathcal{H}_\varepsilon$ on $\Gamma$ with the action $-\varepsilon^{-1}{\mathrm{d}^2/\mathrm{d} x^2}$ on…
The paper deals with some spectral properties of (mostly infinite) quantum and combinatorial graphs. Quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics, nanotechnology, optics, and…
We consider a family of quantum graphs $\{(\Gamma,\mathcal{A}_\varepsilon)\}_{\varepsilon>0}$, where $\Gamma$ is a $\mathbb{Z}^n$-periodic metric graph and the periodic Hamiltonian $\mathcal{A}_\varepsilon$ is defined by the operation…
Quantum graphs are defined by having a Laplacian defined on the edges of a metric graph with boundary conditions on each vertex such that the resulting operator, $\mathbf{L}$, is self-adjoint. We use Neumann boundary conditions although we…
We study a set of scattering matrices of quantum graphs containing minimal number of passbands, i.e., maximal number of zero elements. The cases of even and odd vertex degree are considered. Using a solution of inverse scattering problem,…
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 the spectral gap (the first positive…
We investigate a periodic quantum graph in form of a square lattice with a general self-adjoint coupling at the vertices. We analyze the spectrum, in particular, its high-energy behaviour. Depending on the coupling type, bands and gaps have…
We derive a number of upper and lower bounds for the first nontrivial eigenvalue of a finite quantum graph in terms of the edge connectivity of the graph, i.e., the minimal number of edges which need to be removed to make the graph…
Quantum graphs have recently emerged as models of nonlinear optical, quantum confined systems with exquisite topological sensitivity and the potential for predicting structures with an intrinsic, off-resonance response approaching the…
We identify a set of quantum graphs with unique and precisely defined spectral properties called {\it regular quantum graphs}. Although chaotic in their classical limit with positive topological entropy, regular quantum graphs are…
Motivated by a recent application of quantum graphs to model the anomalous Hall effect we discuss quantum graphs the vertices of which exhibit a preferred orientation. We describe an example of such a vertex coupling and analyze the…
Recently, the work on quantum automorphism groups of graphs has seen renewed progress, which we expand in this paper. Quantum symmetry is a richer notion of symmetry than the classical symmetries of a graph. In general, it is non-trivial to…
We present quantum graphs with remarkably regular spectral characteristics. We call them {\it regular quantum graphs}. Although regular quantum graphs are strongly chaotic in the classical limit, their quantum spectra are explicitly…
We investigate spectral properties of periodic quantum graphs in the form of a kagome or a triangular lattice in the situation when the condition matching the wave functions at the lattice vertices is chosen of a particular form violating…
The symmetries of a finite graph are described by its automorphism group; in the setting of Woronowicz's quantum groups, a notion of a quantum automorphism group has been defined by Banica capturing the quantum symmetries of the graph. In…
Quantum graphs are commonly used as models of complex quantum systems, for example molecules, networks of wires, and states of condensed matter. We consider quantum statistics for indistinguishable spinless particles on a graph,…