On the Essential Spectrum of the Laplacian and Vague Convergence of the Curvature at Infinity

We shall prove that under some volume growth condition, the essential spectrum of the Laplacian contains the interval [(n ? 1)² K/4, ∞) if an n-dimensional Rieman-nian manifold has an end and the average of the part of the Ricci curvature on the end which lies below a nonpositive constant (n ? 1)K converges to zero at infinity.     

二步幂零Lie群上sub-Laplace算子的谱

本文首先研究包括Ｇｒｅｉｎｅｒ算子和二步幂零Ｌｉｅ群上ｓｕｂ Ｌａｐｌａｃｅ算子Ｌ在内的广义齐次线性偏微分算子的谱，然后利用酉表示理论确定了算子Ｌ的谱结构，并深化了Ｋ．Ｆｕｒｕｔａｎｉ，Ｋ．Ｓａｇａｍｉ，Ｎ．Ｏｔｓｕｋｉ关于Ｌａｐｌａｃｅ算子的有关结果．     

Boundary Area Growth and the Spectrum of Discrete Laplacian

We introduce the boundary area growth as a new quantity for an infinite graph. Using this, we give some upper bounds for the bottom of the spectrum of the discrete Laplacian which relates closely to the transition operator. We also give some applications and examples.     

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].     

Twisted Comodule Algebras and the Generalized Quantum Double

In this article we show that the notion of twisted comodule algebras is a generalization of Drinfeld's cocycle twisting construction of bialgebras or Hopf algebras, and that several other twisting constructions, such as Ferrer's twisting product and Lu's one-sided twisting, are also special examples of twisted comodule algebras. Some general properties on twisted comodule algebras are investigated, and applications to the generalized quantum double are given.     

Spectrum of a homogeneous graph

We describe the spectrum of the Laplacian for a homogeneous graph acted on by a discrete group. This follows from a more general result which describes the spectrum of a convolution operator on a homogeneous space of a locally compact group. We also prove a version of Harnack inequality for a Schrödinger operator on an invariant homogeneous graph.     

量子广义Kac-Moody代数的整形式

主要讨论一阶量子广义Kac-Moody代数U_q(2α)的结构,其中a∈Z0.在此基础上,刻画了量子广义代数U_q(g)的另一种整形式.     

On the Laplacian Spectrum and Walk-regular Hypergraphs

We use the generalization of the Laplacian matrix to hypergraphs to obtain several spectral-like results on hypergraphs. For instance, we obtain upper bounds on the eccentricity and the excess of any vertex of hypergraphs. We extend to the case of hypergraphs the concepts of walk regularity and spectral regularity, showing that all walk-regular hypergraphs are spectrally-regular. Finally, we obtain an upper bound on the mean distance of walk-regular hypergraphs that involves all the Laplacian spectrum.     

On the Laplacian Spectrum and Walk-regular Hypergraphs

We use the generalization of the Laplacian matrix to hypergraphs to obtain several spectral-like results on hypergraphs. For instance, we obtain upper bounds on the eccentricity and the excess of any vertex of hypergraphs. We extend to the case of hypergraphs the concepts of walk regularity and spectral regularity, showing that all walk-regular hypergraphs are spectrally-regular. Finally, we obtain an upper bound on the mean distance of walk-regular hypergraphs that involves all the Laplacian spectrum.     

On the Spectrum of the Product of Closed Operators

In this note we study the connection between the spectra of the products AB and BA of unbounded closed operators A and B acting in Banach spaces. Under the condition that the resolvent sets of these products are not empty we show that the spectra of AB and BA coincide away from zero and prove the commutation relation 𝕀 . Further, we prove statements concerning the relationship between the spectra of the operator AB and the block operator matrix .     

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.     

关于四元射影空间的一般极小子流形

本文研究四元射影空间的一般极小子流形,建立了一个积分公式,并且用此公式得到了紧致一般极小子流形的截曲率的Ｐｉｎｃｈｉｎｇ结果．     

极小子流形上Laplace算子的谱

本文讨论了Ｓ^ｎ＋ｐ（１）（或ＣＰ^ｎ＋1）中极小子流形上Ｌａｐｌａｃｅ算子的谱，证明了Ｓ^ｎ＋ｐ（１）中全测地极小子流形（或ＣＰ^ｎ＋1中Ｋａｃｈｌｅｒ超曲面）是由作用在ｑ-形式上的Ｌａｐｌａｃｅ算子的谱唯一确定．     

Quantum Lévy Laplacian and associated heat equation

As a non-commutative extension of the Lévy Laplacian for entire functions on a nuclear space, we define the quantum Lévy Laplacian acting on white noise operators. We solve a heat type equation associated with the quantum Lévy Laplacian and study its relation to the classical Lévy heat equation. The solution to the quantum Lévy heat equation is obtained also from a normal-ordered white noise differential equation involving the quadratic quantum white noise.     

Quantum Mechanics on S 1 Based on Dirac Formalism

We deal with quantum mechanics on S 1 (embedded to R 2 ) based on Dirac formalism. To conclude that the dynamical system is a quantum mechanical system we have to show selfadjointness of the dynamical variables. First we show selfadjointness of the momentum operators and then study their spectra. We find that the continuous spectrum of each momentum operator coincides with the set of all real numbers. Second we discuss selfadjointness of the Hamiltonian for a free quantum mechanical particle moving on S 1 together with its spectrum. We see that the spectrum of the Hamiltonian consists of eigenvalues only. Finally we apply these results to the Cauchy problem for the Schrödinger equation for a free quantum mechanical particle moving on S 1 .     

On the Generalized Ray-Singer Conjecture for Covering Spaces

We find a relation between L² -combinatorial torsion and analytic torsion associated with a small spectrum of the Witten-type Laplacian on a covering space. The result is used as one crucial step in a proof of the generalized Ray-Singer conjecture for covering spaces.     

On the Spectrum of the Resonant Quantum Kicked Rotor

It is proven that none of the bands in the quasi-energy spectrum of the Quantum Kicked Rotor is flat at any primitive resonance of any order. Perturbative estimates of bandwidths at small kick strength are established for the case of primitive resonances of prime order. Different bands scale with different powers of the kicking strength, due to degeneracies in the spectrum of the free rotor.     

一类自伴微分算子谱的离散性

本文研究了２ｎ阶实系数Ｅｕｌｅｒ微分算式生成的对称微分算子,得到了自伴Ｅｕｌｅｒ微分算子的谱是离散的充分必要条件．