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1.
Let be the th Neumann eigenvalue of a bounded domain with piecewisely smooth boundary in . In 1992, P. Kröger proved that , where the upper bound is sharp in view of Weyl's asymptotic formula. The aim of this paper is twofold. First, we will improve this estimate by multiplying a factor in terms of to its right-hand side which approaches strictly from below to 1 as tends to infinity. Second, we will generalize Kröger's estimate to the case when is a compact Euclidean submanifold.
2.
Let Lqr(Ω) be the usual scale of Sobolev spaces and let ΔN be the Neumann Laplacian in an arbitrary Lipschitz domain Ω. We present an interpolation based approach to the following question: for what range of indices does map isomorphically onto Lqr(Ω)/ℝ? 相似文献
3.
《Discrete Mathematics》2022,345(12):113078
Let G be a simple connected graph and let be the sum of the first k largest Laplacian eigenvalues of G. It was conjectured by Brouwer in 2006 that holds for . The case 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 which also confirm the conjecture of Guan et al. in 2014. 相似文献
4.
Rainer Hempel Thomas Kriecherbauer Peter Plankensteiner 《Mathematische Nachrichten》1997,188(1):141-168
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]. 相似文献
5.
Sharp remainder estimates in the Weyl formula for the Neumann Laplacian on a class of planar regions
Yu. Netrusov 《Journal of Functional Analysis》2007,250(1):21-41
We obtain estimates for the counting function of the Neumann Laplacian on a planar domain bounded by the graph of a lower semicontinuous L1-function. These estimates imply necessary and sufficient conditions for the validity of the classical one-term Weyl formula for the counting function and, under certain restrictions, give an order sharp remainder estimate in this formula. 相似文献
6.
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. 相似文献
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设图G是简单连通图.如果任何一个与图G关于拉普拉斯矩阵同谱的图,都与图G同构,称图G可由其拉普拉斯谱确定.定义了树Y_n和树F(2,n,1)两类特殊结构的树.利用同谱图线图的特点,证明了树Y_n和树F(2,n,1)可由其拉普拉斯谱确定. 相似文献
9.
Zoran Stani 《Linear algebra and its applications》2009,431(11):2035-2048
We consider the class of caterpillars with four terminal vertices. Here we prove that every of such caterpillar whose internal path differs in length from both 1 to 3 is uniquely determined by its Laplacian spectrum. Next we take into consideration the remaining two possibilities for the internal path. In the first situation we prove that there is exactly one caterpillar which is not determined by its Laplacian spectrum, while we find an infinite family of such caterpillars in the second. Finally, some observations are given. 相似文献
10.
A. M. Hansson 《偏微分方程通讯》2013,38(12):2157-2163
Let λ k and μ k be the eigenvalues of the Dirichlet and Neumann problems, respectively, in a domain of finite measure in ? d , d > 1. Filonov has proved in a simple way that the inequality μ k+1 < λ k holds for the Laplacian. We extend his result to the Heisenberg Laplacian in three-dimensional domains which fulfill certain geometric conditions. 相似文献
11.
The question of the preservation of discreteness of the spectrum of the Laplacian acting in a space of differential forms under the cutting and gluing of manifolds reduces to the same problem for compact solvability of the operator of exterior derivation. Along these lines, we give some conditions on a cut Y dividing a Riemannian manifold X into two parts X + and X ? under which the spectrum of the Laplacian on X is discrete if and only if so are the spectra of the Laplacians on X + and X ?. 相似文献
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Let H(n; q, n1, n2, n3, n4) be a unicyclic graph with n vertices containing a cycle Cq and four hanging paths Ph1+1, Pn2+1, Pn3+1 and Pn4+1 attached at the same vertex of the cycle. In this paper, it is proved that all unicyclic graphs H (n; q, n1, n2, n3, n4) are determined by their Laplacian spectra. 相似文献
15.
We consider the Laplacian in a straight strip, subject to a combination of Dirichlet and Neumann boundary conditions. We show that a switch of the respective boundary conditions leads to a Hardy inequality for the Laplacian. As a byproduct of our method, we obtain a simple proof of a theorem of Dittrich and Kříž [5]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
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Mirjana Lazić 《Czechoslovak Mathematical Journal》2006,56(4):1207-1213
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. 相似文献
18.
令G是简单图.记L(G)为图G的规范拉普拉斯矩阵,其特征值称为图的规范拉普拉斯特征值.[Adv.Math.(China),2017,46(6):848-856]给出了关于规范拉普拉斯特征值和的相关结论,并提出相关猜想.我们发现在上述文章中的一些重要结果中存在一些错误.本文修正了所有不正确的结果.此外,我们讨论了£(G)的特征值优超不等式.利用这些结果,我们证实了[Adv.Math.(China),2017,46(6):848-8561中提出的一个猜想. 相似文献
19.
Dong Myung Chung Un Cig Ji Kimiaki Saitô 《Proceedings of the American Mathematical Society》2002,130(4):1197-1206
In this paper we shall give some results on a -group generated by the Lévy Laplacian and operators approximating that group in the space of continuous linear operators defined on a certain locally convex space in
20.
N. V. Glotko 《Siberian Mathematical Journal》2008,49(1):18-35
We describe the essential spectrum of the Laplacian over the degree k forms on a class of warped products with a two-dimensional base. 相似文献