Eigenvalues Of Dirac Operator Research Articles

Dependence of eigenvalues of Dirac system on the parameters

AbstractThe dependence of eigenvalues of Dirac system with general boundary conditions is studied. It is shown that the eigenvalues of Dirac operators depend not only continuously but also smoothly on the coefficients, the boundary conditions, and the endpoints of the problem. Furthermore, the differential expressions of the eigenvalues as regards these parameters are given. The results obtained in this paper would provide theoretical support for the numerical calculations of eigenvalues of the corresponding problems.

Scalar flat compactifications of Poincaré-Einstein manifolds and applications

We derive an integral inequality between the mean curvature and the scalar curvature of the boundary of any scalar flat conformal compactifications of Poincaré-Einstein manifolds. As a first consequence, we obtain a sharp lower bound for the first eigenvalue of the conformal half-Laplacian of the boundary of such manifolds. Secondly, a new upper bound for the renormalized volume is given in the four dimensional setting. Finally, some estimates on the first eigenvalues of Dirac operators are also deduced.

Lattice fermions as spectral graphs

We study lattice fermions from the viewpoint of spectral graph theory (SGT). We find that a fermion defined on a certain lattice is identified as a spectral graph. SGT helps us investigate the number of zero eigenvalues of lattice Dirac operators even on the non-torus and non-regular lattice, leading to understanding of the number of fermion species (doublers) on lattices with arbitrary topologies. The procedure of application of SGT to lattice fermions is summarized as follows: (1) One investigates a spectral graph corresponding to a lattice fermion. (2) One obtains a matrix corresponding to the graph. (3) One finds zero eigenvalues of the matrix by use of the discrete Fourier transformation (DFT). (4) By taking an infinite-volume and continuum limits, one finds the number of species. We apply this procedure to the known lattice fermion formulations including Naive fermions, Wilson fermions and Domain-wall fermions, and reproduce the known fact on the number of species. We also apply it to the lattice fermion on the discretized four­dimensional hyperball and discuss the number of fermion species on the bulk. In the end of the paper, we discuss the application of the analysis to lattice fermions on generic lattices with arbitrary topologies, which could lead to constructing a new theorem regarding the number of species.

Mathematical questions about the computation of eigenvalues of Dirac operators with critical potentials in atomic and molecular physics

Cette Note présente divers résultats analytiques et numériques concernant le calcul des valeurs propres de l’opérateur de Dirac, ou plus généralement, des opérateurs avec des “gaps” spectraux. Un algorithme basé sur un théorème abstrait caractérisant les valeurs propres dans les écarts a été trouvé il y a des années, mais ce n’est que récemment qu’une analyse délicate pour identifier et étudier les domaines de ces opérateurs a permis de mettre cet algorithme sur une base ferme concernant le choix des approximations, et ceci aussi bien pour les atomes légers que pour les atomes lourds. Les travaux décrits ici concernent des travaux réalisés avec plusieurs collaborateurs : J. Dolbeault, M. Lewin, M. Loss, E. Séré et M. Vanbreugel.

Location of Eigenvalues of Non-self-adjoint Discrete Dirac Operators

We provide quantitative estimates on the location of eigenvalues of one-dimensional discrete Dirac operators with complex $\ell^p$-potentials for $1\leq p\leq\infty$. As a corollary, subsets of the essential spectrum free of embedded eigenvalues are determined for small $\ell^1$-potential. Further possible improvements and sharpness of the obtained spectral bounds are also discussed.

Kompakt Riemannian Spin-Alt manifold Dirac Operatörünün Alt Sınır Özdeğer Problemleri

In this paper, we construct two modified spinorial LeviCivita connection based on the EnergyMomentum tensor and its trace to bring an optimal lower bound to the eigenvalues of the compact Riemannian Spinsubmanifold Dirac operator in terms of the the EnergyMomentum tensor and its trace. Then, we extend these estimates in terms of the Yamabe number and the area of the submanifold under the conformal change of the metric.

Curious behavior of three-dimensional lattice Dirac operators coupled to a monopole background

We investigate numerically the effect of regulating fermions in the presence of singular background fields in three dimensions. For this, we couple free lattice fermions to a background compact U(1) gauge field consisting of a monopole-anti-monopole pair of magnetic charge $\pm Q$ separated by a distance $s$ in a periodic $L^3$ lattice, and study the low-lying eigenvalues of different lattice Dirac operators under a continuum limit defined by taking $L\to\infty$ at fixed $s/L$. As the background gauge field is parity even, we look for a two-fold degeneracy of the Dirac spectrum that is expected of a continuum-like Dirac operator. The naive-Dirac operator exhibits such a parity-doubling, but breaks the degeneracy of the fermion-doubler modes for the $Q$ lowest eigenvalues in the continuum limit. The Wilson-Dirac operator lifts the fermion-doublers but breaks the parity-doubling in the $Q$ lowest modes even in the continuum limit. The overlap-Dirac operator shows parity-doubling of all the modes even at finite $L$ that is devoid of fermion-doubling, and singles out as a properly regulated continuum Dirac operator in the presence of singular gauge field configurations albeit with a peculiar algorithmic issue.

Eigenvalue estimate of the Dirac operator and Rigidity of Poincare–Einstein metrics

We re-visit the eigenvalue estimate of the Dirac operator on spin manifolds with boundary in terms of the first eigenvalues of conformal Laplace operator as well as the conformal mean curvature operator. These problems were studied earlier by Hijazi-Montiel-Zhang and Raulot and we re-prove them under weaker assumption that a boundary chirality operator exists. Moreover, on these spin manifolds with boundary, we show that any $C^{3,\alpha}$ conformal compactification of some Poincare-Einstein metric must be the standard hemisphere when the first nonzero eigenvalue of the Dirac operator achieves its lowest value, and any $C^{3,\alpha}$ conformal compactification of some Poincare-Einstein metric must be the flat ball in Euclidean space when the first positive eigenvalue of the boundary Dirac operator achieves certain value relating to the second Yamabe invariant.In two cases the Poincare-Einstein metrics are standard hyperbolic metric.

On the number of eigenvalues of the discrete one-dimensional Dirac operator with a complex potential

In this paper we define a one-dimensional discrete Dirac operator on $${\mathbb {Z}}$$ . We study the eigenvalues of the Dirac operator with a complex potential. We obtain bounds on the total number of eigenvalues in the case where V decays exponentially at infinity. We also estimate the number of eigenvalues for the discrete Schrodinger operator with complex potential on $${\mathbb {Z}}$$ . That is we extend the result obtained by Hulko (Bull Math Sci, to appear) to the whole $${\mathbb {Z}}$$ .

Minimally doubled fermions and spontaneous chiral symmetry breaking

Chiral symmetry breaking in massless QCD is a very important feature in the current understanding of low energy physics. Low - lying Dirac modes are suitable to help us understand the spontaneous chiral symmetry breaking, since the formation of a non zero chiral condensate is an effect of their accumulation near zero. The Banks – Casher relation links the spectral density of the Dirac operator to the condensate with an identity that can be read in both directions. In this work we propose a spectral method to achieve a reliable determination of the density of eigenvalues of Dirac operator near zero using the Gauss – Lanczos quadrature. In order to understand better the dynamical chiral symmetry breaking and use the method we propose, we have chosen to work with minimally doubled fermions. These kind of fermions have been proposed as a strictly local discretization of the QCD fermions action, which preserves chiral symmetry at finite cut-off. Being chiral fermions, is easier to work with them and their low - lying Dirac modes and to understand the dynamical spontaneous chiral symmetry breaking.

Schnol’s Theorem and Spectral Properties of Massless Dirac Operators with Scalar Potentials

The spectra of massless Dirac operators are of essential interest e.g. for the electronic properties of graphene, but fundamental questions such as the existence of spectral gaps remain open. We show that the eigenvalues of massless Dirac operators with suitable real-valued potentials lie inside small sets easily characterised in terms of properties of the potentials, and we prove a Schnol'-type theorem relating spectral points to polynomial boundedness of solutions of the Dirac equation. Moreover, we show that, under minimal hypotheses which leave the potential essentially unrestrained in large parts of space, the spectrum of the massless Dirac operator covers the whole real line; in particular, this will be the case if the potential is nearly constant in a sequence of regions.

Dirac spectrum of the Wilson-Dirac operator for QCD with two colors

We study the lattice artefacts of the Wilson Dirac operator for QCD with two colors and fermions in the fundamental representation from the viewpoint of chiral perturbation theory. These effects are studied with the help of the following spectral observables: the level density of the Hermitian Wilson Dirac operator, the distribution of chirality over the real eigenvalues, and the chiral condensate for the quenched as well as for the unquenched theory. We provide analytical expressions for all these quantities. Moreover we derive constraints for the level density of the real eigenvalues of the non-Hermitian Wilson Dirac operator and the number of additional real modes. The latter is a good measure for the strength of lattice artefacts. All computations are confirmed by Monte Carlo simulations of the corresponding random matrix theory which agrees with chiral perturbation theory of two color QCD with Wilson fermions.

Estimates on Complex Eigenvalues for Dirac Operators on the Half-Line

We derive bounds on the location of non-embedded eigenvalues of Dirac operators on the half-line with non-Hermitian L 1-potentials. The results are sharp in the non-relativistic or weak-coupling limit. In the massless case, the absence of discrete spectrum is proved under a smallness assumption.

Eigenvalue Estimates for Non-Selfadjoint Dirac Operators on the Real Line

We show that the non-embedded eigenvalues of the Dirac operator on the real line with non-Hermitian potential $V$ lie in the disjoint union of two disks in the right and left half plane, respectively, provided that the $L^1-norm$ of $V$ is bounded from above by the speed of light times the reduced Planck constant. An analogous result for the Schr\"odinger operator, originally proved by Abramov, Aslanyan and Davies, emerges in the nonrelativistic limit. For massless Dirac operators, the condition on $V$ implies the absence of nonreal eigenvalues. Our results are further generalized to potentials with slower decay at infinity. As an application, we determine bounds on resonances and embedded eigenvalues of Dirac operators with Hermitian dilation-analytic potentials.

The Spinc Dirac operator on hypersurfaces and applications

We extend to the eigenvalues of the hypersurface Spinc Dirac operator well known lower and upper bounds. Examples of limiting cases are then given. Furthermore, we prove a correspondence between the existence of a Spinc Killing spinor on homogeneous 3-dimensional manifolds E⁎(κ,τ) with 4-dimensional isometry group and isometric immersions of E⁎(κ,τ) into the complex space form M4(c) of constant holomorphic sectional curvature 4c, for some c∈R⁎. As applications, we show the non-existence of totally umbilic surfaces in E⁎(κ,τ) and we give necessary and sufficient geometric conditions to immerse a 3-dimensional Sasaki manifold into M4(c).

Lyapunov inequalities and stability for linear Hamiltonian systems

In this paper, we will establish several Lyapunov inequalities for linear Hamiltonian systems, which unite and generalize the most known ones. For planar linear Hamiltonian systems, the connection between Lyapunov inequalities and estimates of eigenvalues of stationary Dirac operators will be given, and some optimal stability criterion will be proved.

Continuity in weak topology: First order linear systems of ODE

In this paper we study important quantities defined from solutions of first order linear systems of ordinary differential equations. It will be proved that many quantities, such as solutions, eigenvalues of one-dimensional Dirac operators, Lyapunov exponents and rotation numbers, depend on the coefficients in a very strong way. That is, they are not only continuous in coefficients with respect to the usual Lp topologies, but also with respect to the weak topologies of the Lp spaces. The continuity results of this paper are a basis to study these quantities in a quantitative way.

Individual eigenvalue distributions of chiral random two-matrix theory and the determination ofFπ

Dirac operator eigenvalues split into two when subjected to two different external vector sources. In a specific finite-volume scaling regime of gauge theories with fermions, this problem can be mapped to a chiral Random Two-Matrix Theory. We derive analytical expressions to leading order in the associated finite-volume expansion, showing how individual Dirac eigenvalue distributions and their correlations equivalently can be computed directly from the effective chiral Lagrangian in the epsilon-regime. Because of its equivalence to chiral Random Two-Matrix Theory, we use the latter for all explicit computations. On the mathematical side, we define and determine gap probabilities and individual eigenvalue distributions in that theory at finite N, and also derive the relevant scaling limit as N is taken to infinity. In particular, the gap probability for one Dirac eigenvalue is given in terms of a new kernel that depends on the external vector source. This expression may give a new and simple way of determining the pion decay constant F_pi from lattice gauge theory simulations.

On the approximation of isolated eigenvalues of ordinary differential operators

We extend a result of Stolz and Weidmann on the approximation of isolated eigenvalues of singular Sturm-Liouville and Dirac operators by the eigenvalues of regular operators.

Variational methods in relativistic quantum mechanics

This review is devoted to the study of stationary solutions of lin- ear and nonlinear equations from relativistic quantum mechanics, involving the Dirac operator. The solutions are found as critical points of an energy func- tional. Contrary to the Laplacian appearing in the equations of nonrelativistic quantum mechanics, the Dirac operator has a negative continuous spectrum which is not bounded from below. This has two main consequences. First, the energy functional is strongly indefinite. Second, the Euler-Lagrange equations are linear or nonlinear eigenvalue problems with eigenvalues lying in a spectral gap (between the negative and positive continuous spectra). Moreover, since we work in the space domain R 3 , the Palais-Smale condition is not satisfied. For these reasons, the problems discussed in this review pose a challenge in the Calculus of Variations. The existence proofs involve sophisticated tools from nonlinear analysis and have required new variational methods which are now applied to other problems. In the first part, we consider the fixed eigenvalue problem for models of a free self-interacting relativistic particle. They allow to describe the localized state of a spin-1/2 particle (a fermion) which propagates without changing its shape. This includes the Soler models, and the Maxwell-Dirac or Klein- Gordon-Dirac equations. The second part is devoted to the presentation of min-max principles al- lowing to characterize and compute the eigenvalues of linear Dirac operators with an external potential, in the gap of their essential spectrum. Many con- sequences of these min-max characterizations are presented, among them a new kind of Hardy-like inequalities and a stable algorithm to compute the eigenvalues. In the third part we look for normalized solutions of nonlinear eigenvalue problems. The eigenvalues are Lagrange multipliers, lying in a spectral gap. We review the results that have been obtained on the Dirac-Fock model which is a nonlinear theory describing the behavior of N interacting electrons in an external electrostatic field. In particular we focus on the problematic definition of the ground state and its nonrelativistic limit. In the last part, we present a more involved relativistic model from Quan- tum Electrodynamics in which the behavior of the vacuum is taken into ac- count, it being coupled to the real particles. The main interesting feature of this model is that the energy functional is now bounded from below, providing us with a good definition of a ground state.