On the spectrum and weakly effective operator for Dirichlet Laplacian in thin deformed tubes

We study the Laplacian in deformed thin (bounded or unbounded) tubes in R³, i.e., tubular regions along a curve r(s) whose cross sections are multiplied by an appropriate deformation function h(s)>0. One of the main requirements on h(s) is that it has a single point of global maximum. We find the asymptotic behaviors of the eigenvalues and weakly effective operators as the diameters of the tubes tend to zero. It is shown that such behaviors are not influenced by some geometric features of the tube, such as curvature, torsion and twisting, and so a huge amount of different deformed tubes are asymptotically described by the same weakly effective operator.     

On the spectrum of the Laplacian

In this article we prove a generalization of Weyl’s criterion for the essential spectrum of a self-adjoint operator on a Hilbert space. We then apply this criterion to the Laplacian on functions over open manifolds and get new results for its essential spectrum.     

On the spectrum of the hierarchical Laplacian

Let (X, d) be a locally compact separable ultrametric space. We assume that (X, d) is proper, that is, any closed ball B?X is a compact set. Given a measure m on X and a function C(B) defined on the set of balls (the choice function) we define the hierarchical Laplacian L _C which is closely related to the concept of the hierarchical lattice of F.J. Dyson. L _C is a non-negative definite self-adjoint operator in L ²(X, m). In this paper we address the following question: How general can be the spectrum \(\mathsf {Spec}(L_{C})\subseteq \mathbb {R}_{+}?\) When (X, d) is compact, S p e c(L _C ) is an increasing sequence of eigenvalues of finite multiplicity which contains 0. Assuming that (X, d) is not compact we show that under some natural conditions concerning the structure of the hierarchical lattice (≡ the tree of d-balls) any given closed subset S ? ?₊, which contains 0 as an accumulation point and is unbounded if X is non-discrete, may appear as S p e c(L _C ) for some appropriately chosen function C(B). The operator ?L _C extends to L ^q (X, m), 1 ≦ q < ∞, as Markov generator and its spectrum does not depend on q. As an example, we consider the operator ?? ^α of fractional derivative defined on the field ? _p of p-adic numbers.     

On the full Brouwer's Laplacian spectrum conjecture

Let G be a simple connected graph and let $S_k(G)$ be the sum of the first k largest Laplacian eigenvalues of G. It was conjectured by Brouwer in 2006 that $S_k(G)\le e(G)+\left(\begin{array}{c}k+1\\ 2\end{array}\right)$ holds for $1\le k\le n?1$. The case $k=2$ was proved by Haemers, Mohammadian and Tayfeh-Rezaie [Linear Algebra Appl., 2010]. In this paper, we propose the full Brouwer's Laplacian spectrum conjecture and we prove the conjecture holds for $k=2$ which also confirm the conjecture of Guan et al. in 2014.     

On the Szeged and the Laplacian Szeged spectrum of a graph

For a given graph G its Szeged weighting is defined by w(e)=n_u(e)n_v(e), where e=uv is an edge of G,n_u(e) is the number of vertices of G closer to u than to v, and n_v(e) is defined analogously. The adjacency matrix of a graph weighted in this way is called its Szeged matrix. In this paper we determine the spectra of Szeged matrices and their Laplacians for several families of graphs. We also present sharp upper and lower bounds on the eigenvalues of Szeged matrices of graphs.     

On the Laplacian spectrum of an infinite graph

In this paper, we introduce the notion of Laplacian spectrum of an infinite countable graph in a different way than in the papers by B. Mohar. We prove some basic properties of this type of spectrum. The approach used is in line with our approach to the limiting spectrum of an infinite graph. The technique of the Laplacian spectrum of finite graphs is essential in this approach.     

On the spectrum of the Dirichlet Laplacian for horn-shaped regions in Rn with infinite volume

An inequality for trace (e^tΔD) is proven, where ?Δ_D is the Dirichlet Laplacian for horn-shaped regions D in Rⁿ. The results of Rozenbljum and Simon for the leading asymptotics for the growth of the number of eigenvalues of the two-dimensional Dirichlet Laplacian in the regions {$\text{(x, y):¦x¦}{}^{\mathrm{\mu }}\text{· ¦y¦ ? 1, μ > 0}$} are easily recovered. An example of a horn-shaped region in R² where that asymptotics is exponential is given.     

Regularity in Capacity and the Dirichlet Laplacian

Given an open set in , we prove that every function in is zero everywhere on the boundary if and only if is regular in capacity. If in addition is bounded, then it is regular in capacity if and only if the mapping from into is injective, where denotes the Perron solution of the Dirichlet problem. Let be the set of all open subsets of which are regular in capacity. Then one can define metrics and on only involving the resolvent of the Dirichlet Laplacian. Convergence in those metrics will be defined to be the local/global uniform convergence of the resolvent of the Dirichlet Laplacian applied to the constant function . We prove that the spaces and are complete and contain the set of all open sets which are regular in the sense of Wiener (or Dirichlet regular) as a closed subset.     

On the reduced signless Laplacian spectrum of a degree maximal graph

For a (simple) graph G, the signless Laplacian of G is the matrix A(G)+D(G), where A(G) is the adjacency matrix and D(G) is the diagonal matrix of vertex degrees of G; the reduced signless Laplacian of G is the matrix Δ(G)+B(G), where B(G) is the reduced adjacency matrix of G and Δ(G) is the diagonal matrix whose diagonal entries are the common degrees for vertices belonging to the same neighborhood equivalence class of G. A graph is said to be (degree) maximal if it is connected and its degree sequence is not majorized by the degree sequence of any other connected graph. For a maximal graph, we obtain a formula for the characteristic polynomial of its reduced signless Laplacian and use the formula to derive a localization result for its reduced signless Laplacian eigenvalues, and to compare the signless Laplacian spectral radii of two well-known maximal graphs. We also obtain a necessary condition for a maximal graph to have maximal signless Laplacian spectral radius among all connected graphs with given numbers of vertices and edges.     

On the growth in a strip of functions defined by series of Dirichlet polynomials

Consideration is given to problems relating to the growth of limit functions of sequences of Dirichlet polynomials in horizontal strips of specific width.Translated from Matematicheskie Zametki, Vol. 3, No. 5, pp. 553–564, May, 1968.     

广义带形区域中的Dirichlet 问题

本文给出广义带形区域中Dirichlet 问题解的积分表示. 如果一类函数在广义带型区域内部调和并在边界上取值为零, 本文给出其需要满足的充要条件.     

On an Approximate Solution of the Dirichlet Problem for the Generalized Laplacian

For an arbitrary differential operator P of order p on an open set X ? R ⁿ, the Laplacian is defined by Δ = P*P. It is an elliptic differential operator of order 2p provided the symbol mapping of P is injective. Let O be a relatively compact domain in X with smooth boundary, and B_j(j = 0…,p — 1) be a Dirichlet system of order p ? 1 on ?O. By {C_j} we denote the Dirichlet system on ?O adjoint for {B_j} with respect to the Green formula for P. The Hardy space H²(O) is defined to consist of all the solutions f of Δf = 0 in O of finite order of growth near the boundary such that the weak boundary values of the expression {B_jf} and {C_j(Pf)} belong to the Lebesgue space L²(?O). Then the Dirichlet problem consists of finding a solution f ? H²(O) with prescribed data {B_jf} on ?O. We develop the classical Fischer-Riesz equations method to derive a solvability condition of the Dirichlet problem as well as an approximate formula for solutions.     

On periodic groups with narrow spectrum

We study groups with no elements of big orders. We prove that if the set of element orders of G is {1, 2, 3, 4, p, 9}, where p ∈ {7, 5}, then G is locally finite.     

Limiting absorption principle for the magnetic Dirichlet Laplacian in a half-plane

We consider the Dirichlet Laplacian in the half-plane with a constant magnetic field. Due to the translational invariance, this operator admits a fiber decomposition and a family of dispersion curves that are real analytic functions. Each of them is simple and monotonically decreasing from positive infinity to a finite value, which is the corresponding Landau level. These finite limits are thresholds in the purely absolutely continuous spectrum of the magnetic Laplacian. We prove a limiting absorption principle for this operator, both outside and at the thresholds. Finally, we establish analytic and decay properties for functions lying in the absorption spaces. We point out that the analysis carried out in this article is rather general, and can be adapted to a wide class of fibered magnetic Laplacians with thresholds in their spectrum that are finite limits of their band functions.