No Arabic abstract
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 of the corresponding Laplacian into the orthogonal sum of Sturm--Liouville operators. In contrast to the case of radially symmetric trees, the deficiency indices of the Laplacian defined on the minimal domain are at most one and they are equal to one exactly when the corresponding metric antitree has finite total volume. In this case, we provide an explicit description of all self-adjoint extensions including the Friedrichs extension. Furthermore, using the spectral theory of Krein strings, we perform a thorough spectral analysis of this model. In particular, we obtain discreteness and trace class criteria, criterion for the Kirchhoff Laplacian to be uniformly positive and provide spectral gap estimates. We show that the absolutely continuous spectrum is in a certain sense a rare event, however, we also present several classes of antitrees such that the absolutely continuous spectrum of the corresponding Laplacian is $[0,infty)$.
We construct rich families of Schrodinger operators on symmetric graphs, both quantum and combinatorial, whose spectral degeneracies are persistently larger than the maximal dimension of an irreducible representations of the symmetry group.
In geometric analysis, an index theorem relates the difference of the numbers of solutions of two differential equations to the topological structure of the manifold or bundle concerned, sometimes using the heat kernels of two higher-order differential operators as an intermediary. In this paper, the case of quantum graphs is addressed. A quantum graph is a graph considered as a (singular) one-dimensional variety and equipped with a second-order differential Hamiltonian H (a Laplacian) with suitable conditions at vertices. For the case of scale-invariant vertex conditions (i.e., conditions that do not mix the values of functions and of their derivatives), the constant term of the heat-kernel expansion is shown to be proportional to the trace of the internal scattering matrix of the graph. This observation is placed into the index-theory context by factoring the Laplacian into two first-order operators, H =A*A, and relating the constant term to the index of A. An independent consideration provides an index formula for any differential operator on a finite quantum graph in terms of the vertex conditions. It is found also that the algebraic multiplicity of 0 as a root of the secular determinant of H is the sum of the nullities of A and A*.
In this paper, we study eigenvalues and eigenfunctions of $p$-Laplacians with Dirichlet boundary condition on graphs. We characterize the first eigenfunction (and the maximum eigenfunction for a bipartite graph) via the sign condition. By the uniqueness of the first eigenfunction of $p$-Laplacian, as $pto 1,$ we identify the Cheeger constant of a symmetric graph with that of the quotient graph. By this approach, we calculate various Cheeger constants of spherically symmetric graphs.
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 its edges; here $varepsilon>0$ is a small parameter. Let $minmathbb{N}$. We show that under a proper choice of vertex conditions the spectrum $sigma(mathcal{H}^varepsilon)$ of $mathcal{H}^varepsilon$ has at least $m$ gaps as $varepsilon$ is small enough. We demonstrate that the asymptotic behavior of these gaps and the asymptotic behavior of the bottom of $sigma(mathcal{H}^varepsilon)$ as $varepsilonto 0$ can be completely controlled through a suitable choice of coupling constants standing in those vertex conditions. We also show how to ensure for fixed (small enough) $varepsilon$ the precise coincidence of the left endpoints of the first $m$ spectral gaps with predefined numbers.
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.