Related papers: Sharp bounds for the largest eigenvalue

We give estimates for the $L^p$ norm ($2\leq p \leq +\infty$) of the restriction to a curve of the eigenfunctions of the Laplace Beltrami operator on a Riemannian surface. If the curve is a geodesic, we show that on the sphere these…

We obtain upper bounds for the Steklov eigenvalues of warped products $\Omega\times_h\Sigma$, where $\Omega$ is a compact Riemannian manifold with boundary and $\Sigma$ is a closed Riemannian manifold. These bounds involve the volume of…

For a simple and connected graph, several lower and upper bounds of graph invariants expressed in terms of the eigenvalues of the normalized Laplacian matrix have been proposed in literature. In this paper, through a unified approach based…

Let $M$ denote a compact, connected Riemannian manifold of dimension $n\in{\mathbb N}$. We assume that $ M$ has a smooth and connected boundary. Denote by $g$ and ${\rm d}v_g$ respectively, the Riemannian metric on $M$ and the associated…

For a self--adjoint Laplace operator on a finite, not necessarily compact, metric graph lower and upper bounds on each of the negative eigenvalues are derived. For compact finite metric graphs Poincar\'{e} type inequalities are given.

We consider the question of giving an upper bound for the first nontrivial eigenvalue of the Wentzell-Laplace operator of a domain $\Omega$, involving only geometrical informations. We provide such an upper bound, by generalizing Brock's…

We present sharp inequalities relating the number of vertices, edges, and triangles of a graph to the smallest eigenvalue of its adjacency matrix and the largest eigenvalue of its Laplacian.

In 1993 Hong asked what are the best bounds on the $k$'th largest eigenvalue $\lambda_{k}(G)$ of a graph $G$ of order $n$. This challenging question has never been tackled for any $2<k<n$. In the present paper tight bounds are obtained for…

We study the high energy estimate for the resolvent $R(\lambda)$ of the Laplacian on non-trapping asymptotically hyperbolic manifolds (AHM). In the literature, polynomial bound of the form $\|R(\lambda)\| = O(|\lambda|^{N})$ for $|\lambda|$…

Eigenvalues of a graph are the eigenvalues of the corresponding (0,1)-adjacency matrix. The second largest eigenvalue lambda_2 provides significant information on characteristics and structure of graphs. Therefore, finding bounds for…

A theorem of J. Hersch (1970) states that for any smooth metric on $S^2$, with total area equal to $4\pi$, the first nonzero eigenvalue of the Laplace operator acting on functions is less than or equal to 2 (this being the value for the…

Let G be a random subgraph of the n-cube where each edge appears randomly and independently with probability p. We prove that the largest eigenvalue of the adjacency matrix of G is almost surely \lambda_1(G)= (1+o(1))…

We study $L^p$ bounds on spectral projections for the Laplace operator on compact Riemannian manifolds, restricted to small frequency dependent neighborhoods of submanifolds. In particular, if $\lambda$ is a frequency and the size of the…

In this paper, we investigate on a bounded open set of $\mathbb{R}^N$ with smooth boundary, an eigenvalue problem involving the sum of nonlocal operators $(-\Delta)_p^{s_1}+ (-\Delta)_q^{s_2}$ with $s_1,s_2\in (0,1)$, $p,q\in (1,\infty)$…

In this paper, we obtain the bounds of the extreme eigenvalues of a normalized and signless Laplacian matrices using by their traces. In addition, we determine the bounds for k-th eigenvalues of normalized and signless Laplacian matrices.

We study the existence and properties of metrics maximising the first Laplace eigenvalue among conformal metrics of unit volume on Riemannian surfaces. We describe a general approach to this problem and its higher eigenvalue versions via…

We prove pointwise bounds for $L^2$ eigenfunctions of the Laplace-Beltrami operator on locally symmetric spaces with $\mathbb{Q}$-rank one if the corresponding eigenvalues lie below the continuous part of the $L^2$ spectrum. Furthermore, we…

Let $\Omega\subset\mathbb{R}^n$ be an open set with the same volume as the unit ball $B$ and let $\lambda_k(\Omega)$ be the $k$-th eigenvalue of the Laplace operator of $\Omega$ with Dirichlet boundary conditions on $\partial\Omega$. In…

The known upper bounds for the multiplicities of the Laplace-Beltrami operator eigenvalues on the real projective plane are improved for the eigenvalues with even indexes. Upper bounds for Dirichlet, Neumann and Steklov eigenvalues on the…

In this paper we prove the optimal upper bound $\frac{\lambda_{n}}{\lambda_{m}}\leq\frac{n^{2}}{m^{2}}$ $\Big(\lambda_{n}>\lambda_{m}\geq 11\sup\limits_{x\in[0,1]}q(x)\Big)$ for one-dimensional Schrodinger operators with a nonnegative…