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Prescribing eigenvalues of the Dirac operator  [PDF]
Mattias Dahl
Mathematics , 2003, DOI: 10.1007/s00229-005-0583-0
Abstract: In this note we show that every compact spin manifold of dimension $\geq 3$ can be given a Riemannian metric for which a finite part of the spectrum of the Dirac operator consists of arbitrarily prescribed eigenvalues with multiplicity 1.
Absence of eigenvalues for the generalized two-dimensional periodic Dirac operator  [PDF]
L. I. Danilov
Mathematics , 2007,
Abstract: A generalized two-dimensional periodic Dirac operator is considered, with $L^{\infty}$-matrix-valued coefficients of the first order derivatives and with complex matrix-valued potential. It is proved that if the matrix-valued potential has zero bound relative to the free Dirac operator, then the spectrum of the operator in question contains no eigenvalues.
A comparison of the eigenvalues of the Dirac and Laplace operator on the two-dimensional torus  [PDF]
Ilka Agricola,Bernd Ammann,Thomas Friedrich
Mathematics , 1998,
Abstract: We compare the eigenvalues of the Dirac and Laplace operator on a two-dimensional torus with respect to the trivial spin structure. In particular, we compute their variation up to order 4 upon deformation of the flat metric, study the corresponding Hamiltonian and discuss several families of examples.
Dirac Operator and Eigenvalues in Riemannian Geometry  [PDF]
Giampiero Esposito
Physics , 1995,
Abstract: The aim of the lectures is to introduce first-year Ph.D. students and research workers to the theory of the Dirac operator, spinor techniques, and their relevance for the theory of eigenvalues in Riemannian geometry. Topics: differential operators on manifolds, index of elliptic operators, Dirac operator, index problem for manifolds with a boundary, index of the Dirac operator and anomalies, spectral asymmetry and Riemannian geometry, spectral or local boundary conditions for massless spin-1/2 fields, potentials for massless spin-3/2 fields, conformal anomalies for massless spin-1/2 fields.
Extrinsic Bounds for Eigenvalues of the Dirac Operator  [PDF]
Christian Baer
Mathematics , 1998,
Abstract: 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.
The Polyakov Loop and the Eigenvalues of the Dirac Operator  [PDF]
Wolfgang S?ldner
Physics , 2007,
Abstract: Aiming at the link between confinement and chiral symmetry the Polyakov loop represented as a spectral sum of eigenvalues of the Dirac operator was subject of recent studies. We analyze the volume dependence as well as the continuum behavior of this quantity for quenched QCD using staggered fermions. Furthermore, we present first results using dynamical configurations.
Eigenvalues of the Dirac Operator on Manifolds with Boundary  [PDF]
Oussama Hijazi,Sebastian Montiel,Xiao Zhang
Physics , 2000, DOI: 10.1007/s002200100475
Abstract: Under standard local boundary conditions or certain global APS boundary conditions, we get lower bounds for the eigenvalues of the Dirac operator on compact spin manifolds with boundary. Limiting cases are characterized by the existence of real Killing spinors and the minimality of the boundary.
On the eigenvalues of the twisted Dirac operator  [PDF]
Marcos Jardim Rafael F. Le?o
Mathematics , 2008,
Abstract: Given a compact Riemannian spin manifold with positive scalar curvature, we find a family of connections $\nabla^{A_t}$ for $t\in[0,1]$ on a trivial vector bundle of sufficiently high rank, such that the first eigenvalue of the twisted Dirac operator $D_{A_t}$ is nonzero and becomes arbitrarily small as $t\to1$. However, if one restricts the class of twisting connections considered, then nonzero lower bounds do exist. We illustrate this fact by establishing a nonzero lower bound for the Dirac operator twisted by Hermitian-Einstein connections over Riemann surfaces.
Distributions of Dirac Operator Eigenvalues  [PDF]
G. Akemann,P. H. Damgaard
Physics , 2003, DOI: 10.1016/j.physletb.2003.12.024
Abstract: The distribution of individual Dirac eigenvalues is derived by relating them to the density and higher eigenvalue correlation functions. The relations are general and hold for any gauge theory coupled to fermions under certain conditions which are stated. As a special case, we give examples of the lowest-lying eigenvalue distributions for QCD-like gauge theories without making use of earlier results based on the relation to Random Matrix Theory.
Optimal eigenvalues estimate for the Dirac operator on domains with boundary  [PDF]
Simon Raulot
Mathematics , 2006, DOI: 10.1007/s11005-005-0005-y
Abstract: We give a lower bound for the eigenvalues of the Dirac operator on a compact domain of a Riemannian spin manifold under the $\MIT$ bag boundary condition. The limiting case is characterized by the existence of an imaginary Killing spinor.
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