Lower Bounds for the Eigenvalues of the Dirac Operator: Part I. The Hypersurface Dirac Operator

We give optimal lower bounds for the hypersurface Diracoperator in terms of the Yamabe number, the energy-momentum tensor andthe mean curvature. In the limiting case, we prove that the hypersurfaceis an Einstein manifold with constant mean curvature.     

Extrinsic Bounds for Eigenvalues of the Dirac Operator

We derive upper eigenvalue bounds for the Dirac operator of a closed hypersurface in a manifold with Killing spinors such as Euclidean space, spheres or hyperbolic space. The bounds involve the Willmore functional. Relations with the Willmore inequality are briefly discussed. In higher codimension we obtain bounds on the eigenvalues of the Dirac operator of the submanifold twisted with the spinor bundle of the normal bundle.     

Eigenvalues of the Dirac Operator on Fibrations over S1

We consider the Dirac operator on fibrations overS ¹ which have up to holonomy a warped product metric. Wegive lower bounds for the eigenvalues on M and if the Diracoperator on the typical fibre F has a kernel, we calculatethe corresponding part of the spectrum on M explicitly.Moreover, we discuss the dependence of the spectrum of theholonomy and obtain bounds for the multiplicity of the eigenvalues.     

Eigenvalues of Spinc Dirac Operator

In this paper we research the lower bound of the eigenvalue of Spinc Dirac operator on the Spinc manifold. By the Weisenbock formula, we get an estimate of it, then following the idea of Th Friedrich [2] and X Zhang [6]. We get a finer estimate of it. As an application, we give a condition when the Seiberg-Witten equation only has 0 solution.     

On the Absolutely Continuous Spectrum of Dirac Operator

Abstract

We prove that the massless Dirac operator in ?³ with long-range potential has an a.c. spectrum which fills the whole real line. The Dirac operators with matrix-valued potentials are considered as well.     

Dirac Eigenvalues for Generic Metrics on Three-Manifolds

We show that for generic Riemannian metrics on a closed spin manifold of dimension three the Dirac operator has only simple eigenvalues.     

Inequalities of Eigenvalues for the Dirac Operator on Compact Complex Spin Submanifolds in Complex Projective Spaces

For a compact complex spin manifold M with a holomorphic isometric embed- ding into the complex projective space,the authors obtain the extrinsic estimates from above and below for eigenvalues of the Dirac operator,which depend on the data of an isometric embedding of M.Further,from the inequalities of eigenvalues,the gaps of the eigenvalues and the ratio of the eigenvalues are obtained.     

Functions of the Laplace Operator on Manifolds with Lower Ricci and Injectivity Bounds

Recent work of G. Mauceri, S. Meda, and M. Vallarino produces L ^p estimates on a natural class of functions of the Laplace–Beltrami operator on a Riemannian manifold M, under fairly weak geometrical hypotheses, namely lower bounds on its injectivity radius and Ricci tensor, but with an auxiliary decay hypothesis on the heat semigroup. We sharpen this result by removing the decay hypothesis.     

双调和算子本征值的上界

设Ω是 Rn中的有界区域 ,其边界足够光滑 ,λk为双调和算子在自由边界条件下的第 k个本征值 ,利用变分原理及 Fourier变换 ,给出了本征值部分和 ∑kj=1λj的一个上界 ,该上界仅依赖于区域的体积 .     

The Dirac spectrum on manifolds with gradient conformal vector fields

We show that the Dirac operator on a spin manifold does not admit L² eigenspinors provided the metric has a certain asymptotic behaviour and is a warped product near infinity. These conditions on the metric are fulfilled in particular if the manifold is complete and carries a non-complete vector field which outside a compact set is gradient conformal and non-vanishing.     

Twistor Operators and Eigenvalues of the Dirac Operator on Compact Quaternion-Kähler Spin Manifolds

Quaternion-Kähler twistor operators are introduced. Using these operators with the Lichnerowicz formula, we get lower bounds for the square of the eigenvalues of the Dirac operator in terms of the eigenvalues of the fundamental 4-form.     

Conformally invariant powers of the Dirac operator in Clifford analysis

The paper deals with conformally invariant higher‐order operators acting on spinor‐valued functions, such that their symbols are given by powers of the Dirac operator. A general classification result proves that these are unique, up to a constant multiple. A general construction for such an invariant operators on manifolds with a given conformal spin structure was described in (Conformally Invariant Powers of the Ambient Dirac Operator. ArXiv math.DG/0112033, preprint), generalizing the case of powers of the Laplace operator from (J. London Math. Soc. 1992; 46 :557–565). Although there is no hope to obtain explicit formulae for higher powers of the Laplace or Dirac operator on a general manifold, it is possible to write down an explicit formula on Einstein manifolds in case of the Laplace operator (see Laplacian Operators and Curvature on Conformally Einstein Manifolds. ArXiv: math/0506037, 2006). Here we shall treat the spinor case on the sphere. We shall compute the explicit form of such operators on the sphere, and we shall show that they coincide with operators studied in (J. Four. Anal. Appl. 2002; 8 (6):535–563). The methods used are coming from representation theory combined with traditional Clifford analysis techniques. Copyright © 2010 John Wiley & Sons, Ltd.     

Semi-Classical Green Kernel Asymptotics for the Dirac Operator

We consider a semi-classical Dirac operator in d ∈ ? spatial dimensions with a smooth potential whose partial derivatives of any order are bounded by suitable constants. We prove that the distribution kernel of the inverse operator evaluated at two distinct points fulfilling a certain hypothesis can be represented as the product of an exponentially decaying factor involving an associated Agmon distance and some amplitude admitting a complete asymptotic expansion in powers of the semi-classical parameter. Moreover, we find an explicit formula for the leading term in that expansion.     

On the Kernel of the Equivariant Dirac Operator

We prove a vanishing theorem for certain isotypical components of the kernel of the S¹-equivariant Dirac operator with coefficients in an admissible Clifford module. The method is based on changing the metric by a conformal (generally unbounded) factor and studying the effect of this change on the Dirac operator and its kernel. In the cases relevant to S¹-actions we find that the kernel of the new operator is naturally isomorphic to the kernel of the original operator.     

高阶椭圆算子Dirichlet本征值的下界

设Ω是R^m(m≥2）中的一个有界区域,其边界足够光滑,考察2p(p≥1）阶椭圆算子（－1）^p ∑│α│＝│β│＝pa^α(Aαβa^β)在Dirichlet边界条件下的本征值问题,给出了其本征值的一个下界,该下界除与维数m有关外仅依赖于区域Ω的体积。     

On the contribution of the Coulomb singularity of arbitrary charge to the Dirac Hamiltonian

We study the self-adjoint extensions of the Dirac operator , where the electrical potential contains a Coulomb singularity of arbitrary charge and the magnetic potential is allowed to be unbounded at infinity. We show that if the Coulomb singularity has the form where has a limit at 0, then, for any self-adjoint extension of the Dirac operator, removing the singularity results in a Hilbert-Schmidt perturbation of its resolvent.

     

A Sobolev-like inequality for the Dirac operator

In this article, we prove a Sobolev-like inequality for the Dirac operator on closed compact Riemannian spin manifolds with a nearly optimal Sobolev constant. As an application, we give a criterion for the existence of solutions to a nonlinear equation with critical Sobolev exponent involving the Dirac operator. We finally specify a case where this equation can be solved.     

The Spectrum of the Dirac Operator on G2/SO(4)

The spectrum of the Dirac operator on the compact symmetric space G²/SO(4) is calculated using Parthasarathy's formula and classical harmonic analysis. Furthermore, the neccessary branching rules for G² and its subgroup SO(4) are given.     

Dirac方程的特征函数的迹公式

本文研究具有周期有限带位势的Dirac算子, 利用Dirac算子与单值算子的交换性,定义Bloch函数和乘子曲线, 进而获得揭示Bloch函数与位势间内在关系的Dubrovin-Novikov型公式; 通过复球面上的留数计算, 最终分别得到相应于谱带左端点、右端点以及双侧端点的特征函数的迹公式.     

An approach to the spectrum structure of Dirac operators by the local-compactness method

The purpose of this paper is to investigate the spectra of the Dirac operator . The local compactness of is shown under some assumption on . This method enables us to prove that if as , then and to give a significant sufficient condition that or has a purely discrete spectrum.