Tubes and eigenvalues for negatively curved manifolds

We investigate the structure of the spectrum near zero for the Laplace operator on a complete negatively curved Riemannian manifoldM. If the manifold is compact and its sectional curvaturesK satisfy 1 ≤K < 0, we show that the smallest positive eigenvalue of the Laplacian is bounded below by a constant depending only on the volume ofM. Our result for a complete manifold of finite volume with sectional curvatures pinched between −a² and −1 asserts that the number of eigenvalues of the Laplacian between 0 and (n− 1)²/4 is bounded by a constant multiple of the volume of the manifold with the constant depending ona and the dimension only. Research supported in part by the Swiss National Science Foundation, the US National Science Foundation, and the PSC-CUNY Research Award Program.     

Small eigenvalues on p-forms for collapsings of the even-dimensional spheres

 We prove that for some p there exist small eigenvalues of the Laplacian on p-forms for collapsings of the even dimensional spheres with curvature bounded below. These collapsings were constructed by T. Yamaguchi. Received: 14 January 2002 / Revised version 25 April 2002     

Regular approximations of singular Sturm-Liouville problems

Given any self-adjoint realization S of a singular Sturm-Liouville (S-L) problem, it is possible to construct a sequence {S_r{ of regular S-L problems with the properties (i) every point of the spectrum of S is the limit of a sequence of eigenvalues from the spectrum of the individual members of {S_r{ (ii) in the case when 5 is regular or limit-circle at each endpoint, a convergent sequence of eigenvalues from the individual members of {S_r{ has to converge to an eigenvalue of S (iii) in the general case when S is bounded below, property (ii) holds for all eigenvalues below the essential spectrum of S.     

The distribution of the eigenvalues of the discrete Laplacian

We estimate the distribution of the eigenvalues of the discrete Laplacian on a bounded set inR ⁿ . The proof is based on a variational technique similar to that used by Weyl for the Laplacian. As an application of our estimates we prove stability in the maximum norm for the Crank-Nicolson method for the heat equation on a bounded set.The research of the second author was supported in part by National Science Foundation grant GP-30735X.     

Approximation of approximation numbers by truncation

LetA be a bounded linear operator onsome infinite-dimensional separable Hilbert spaceH and letA _n be the orthogonal compression ofA to the span of the firstn elements of an orthonormal basis ofH. We show that, for eachk1, the approximation numberss _k(A_n) converge to the corresponding approximation numbers _k(A) asn. This observation implies almost at once some well known results on the spectral approximation of bounded selfadjoint operators. For example, it allows us to identify the limits of all upper and lower eigenvalues ofA _n in the case whereA is selfadjoint. These limits give us all points of the spectrum of a selfadjoint operator which lie outside the convex hull of the essential spectrum. Moreover, it follows that the spectrum of a selfadjoint operatorA with a connected essential spectrum can be completely recovered from the eigenvalues ofA _n asn goes to infinity.     

Examples of embedded eigenvalues for the Dirichlet Laplacian in perturbed waveguides

It is shown that there exist domains Ω ? ?^N, which outside of some ball coincide with the strip ?^{N ? 1} × (0, π) and for which the Dirichlet Laplacian – Δ has eigenvalues within the subinterval (1, 4) of the essential spectrum (1, ∞).     

Sur le spectre des fibrés en tore qui s'effondrent

 We consider torus bundles over S ¹ and T ² with solmanifold structure, and analyze the behavior of the Laplacian acting on left-invariant differantial forms under homogeneous collapsings with bounded diameter and bounded sectional curvature. Whe show how the number of small eigenvalues depends on the topology of the bundle and the geometry of the collapsing.

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Sur le spectre des fibrés en tore qui s'effondrent

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Received: 14 January 2002     

On graphs whose Laplacian matrices have distinct integer eigenvalues

In this paper, we investigate graphs for which the corresponding Laplacian matrix has distinct integer eigenvalues. We define the set S_i,n to be the set of all integers from 0 to n, excluding i. If there exists a graph whose Laplacian matrix has this set as its eigenvalues, we say that this set is Laplacian realizable. We investigate the sets S_i,n that are Laplacian realizable, and the structures of the graphs whose Laplacian matrix has such a set as its eigenvalues. We characterize those i < n such that S_i,n is Laplacian realizable, and show that for certain values of i, the set S_i,n is realized by a unique graph. Finally, we conjecture that S_n,n is not Laplacian realizable for n ≥ 2 and show that the conjecture holds for certain values of n. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Discrete and Cantor Spectrum for Neumann Laplacians of Combs

Extending results of [HSS] on the construction of Neumann Laplacians of combs with given essential spectrum S = S [0,∞], we show that, in addition to the essential spectrum, a bounded sequence of discrete eigenvalues can be prescribed. More generally, we find that certain types of spectra can be constructed precisely, in a bounded interval [0, s].     

The radial curvature of an end that makes eigenvalues vanish in the essential spectrum I

The concern of this paper is to clarify a relationship between the curvatures at infinity and the spectral structure of the Laplacian. In particular, this paper discusses the question of whether there is an eigenvalue of the Laplacian embedded in the essential spectrum or not. The borderline-behavior of the radial curvatures for this problem will be determined: we will assume that the radial curvature K _rad. of an end converges to a constant −1 at infinity with the decay order K _rad. + 1 = o(r ⁻¹) and prove the absence of eigenvalues embedded in the essential spectrum. Furthermore, in order to show that this decay order K _rad. + 1 = o(r ⁻¹) is sharp, we will construct a manifold with the radial curvature decay K _rad. + 1 = O(r ⁻¹) and with an eigenvalue \frac(n-1)²4+1{\frac{(n-1)^2}{4}+1} embedded in the essential spectrum [ \frac(n-1)²4, ￥){[ \frac{(n-1)^2}{4}, \infty)} of the Laplacian.     

Spectral stability of the Robin Laplacian

We consider the Robin Laplacian in two bounded regions Ω₁ and Ω₂ of ℝ ^N with Lipschitz boundaries and such that Ω₂ ⊂ Ω₁, and we obtain two-sided estimates for the eigenvalues λ _n,2 of the Robin Laplacian in Ω₂ via the eigenvalues λ _{n, 1} of the Robin Laplacian in Ω₁. Our estimates depend on the measure of the set difference Ω\Ω₂ and on suitably defined characteristics of vicinity of the boundaries ∂Ω₁ and ∂Ω₂, and of the functions defined on ∂Ω₁ and on ∂Ω₂ that enter the Robin boundary conditions.     

Eigenvalue estimate for the rough Laplacian on differential forms

In this article, we study the spectrum of the rough Laplacian acting on differential forms on a compact Riemannian manifold (M, g). We first construct on M metrics of volume 1 whose spectrum is as large as desired. Then, provided that the Ricci curvature of g is bounded below, we relate the spectrum of the rough Laplacian on 1-forms to the spectrum of the Laplacian on functions, and derive some upper bound in agreement with the asymptotic Weyl law.     

Spectral Convergence of Quasi-One-Dimensional Spaces

We consider a family of non-compact manifolds X_ε (“graph-like manifolds”) approaching a metric graph X₀ and establish convergence results of the related natural operators, namely the (Neumann) Laplacian and the generalized Neumann (Kirchhoff) Laplacian on the metric graph. In particular, we show the norm convergence of the resolvents, spectral projections and eigenfunctions. As a consequence, the essential and the discrete spectrum converge as well. Neither the manifolds nor the metric graph need to be compact, we only need some natural uniformity assumptions. We provide examples of manifolds having spectral gaps in the essential spectrum, discrete eigenvalues in the gaps or even manifolds approaching a fractal spectrum. The convergence results will be given in a completely abstract setting dealing with operators acting in different spaces, applicable also in other geometric situations. Communicated by Claude Alain Pillet Submitted: December 21, 2005 Accepted: January 30, 2006     

On diagrammatic bounds of knot volumes and spectral invariants

In recent years, several families of hyperbolic knots have been shown to have both volume and λ₁ (first eigenvalue of the Laplacian) bounded in terms of the twist number of a diagram, while other families of knots have volume bounded by a generalized twist number. We show that for general knots, neither the twist number nor the generalized twist number of a diagram can provide two-sided bounds on either the volume or λ₁. We do so by studying the geometry of a family of hyperbolic knots that we call double coil knots, and finding two-sided bounds in terms of the knot diagrams on both the volume and on λ₁. We also extend a result of Lackenby to show that a collection of double coil knot complements forms an expanding family iff their volume is bounded.     

‘Universal’ inequalities for the eigenvalues of the Hodge de Rham Laplacian

We obtain ‘universal’ inequalities for the eigenvalues of the Laplacian acting on differential forms of a Euclidean compact submanifold. These inequalities generalize the Yang inequality concerning the eigenvalues of the Dirichlet Laplacian of a bounded Euclidean domain.      

On the Laplacian energy of a graph

In this paper we consider the energy of a simple graph with respect to its Laplacian eigenvalues, and prove some basic properties of this energy. In particular, we find the minimal value of this energy in the class of all connected graphs on n vertices (n = 1, 2, ...). Besides, we consider the class of all connected graphs whose Laplacian energy is uniformly bounded by a constant α ⩾ 4, and completely describe this class in the case α = 40.     

Comparison theorems on Finsler manifolds with weighted Ricci curvature bounded below

We obtain the Laplacian comparison theorem and the Bishop-Gromov comparison theorem on a Finsler manifold with the weighted Ricci curvature Ric_∞ bounded below. As applications, we prove that if the weighted Ricci curvature Ric_∞ is bounded below by a positive number, then the manifold must have finite fundamental group, and must be compact if the distortion is also bounded. Moreover, we give the Calabi-Yau linear volume growth theorem on a Finsler manifold with nonnegative weighted Ricci curvature.     

On the Uniform Poincaré Inequality

We give a proof of the Poincaré inequality in W ^{1, p} (Ω) with a constant that is independent of Ω ? , where  is a set of uniformly bounded and uniformly Lipschitz domains in ? ⁿ . As a byproduct, we obtain the following: The first non vanishing eigenvalues λ₂(Ω) of the standard Neumann (variational) boundary value problem on Ω for the Laplace operator are bounded below by a positive constant if the domains Ω vary and remain uniformly bounded and uniformly Lipschitz regular.     

On the distribution of Laplacian eigenvalues of a graph

This paper presents some bounds on the number of Laplacian eigenvalues contained in various subintervals of [0, n] by using the matching number and edge covering number for G, and asserts that for a connected graph the Laplacian eigenvalue 1 appears with certain multiplicity. Furthermore, as an application of our result (Theorem 13), Grone and Merris’ conjecture [The Laplacian spectrum of graph II. SIAM J. Discrete Math., 7, 221–229 (1994)] is partially proved.