On the point spectrum of Dirac operators

Berthier, A. M.; Georgescu, Vladimir

doi:10.1016/0022-1236(87)90007-3

Cited by 60 publications

(71 citation statements)

References 4 publications

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“…It suffices to put together Proposition 2.2, Theorem 2.11 and Corollary 2.12. K Results on the absence of eigenvalues embedded in the continuous spectrum of H can be found in [1], but we do not need them.…”

Section: Lemma 28 Suppose That Functions V and Amentioning

confidence: 97%

Scattering Theory for the Dirac Operator with a Long-Range Electromagnetic Potential

Gâtel

Yafaev

2001

Journal of Functional Analysis

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We consider the Dirac operator with a long-range potential V(x). Scalar, pseudoscalar and vector components of V(x) may have arbitrary power-like decay at infinity. We introduce wave operators with time-independent modifiers. These modifiers are pseudo-differential operators whose symbols are, roughly speaking, constructed in terms of approximate eigenfunctions of the stationary problem. We derive and solve eikonal and transport equations for the corresponding phase and amplitude functions. From an analytical point of view, our proof of the existence and completeness of the wave operators relies on the limiting absorption principle and radiation estimates established in the paper. This allows us to fit the long-range scattering theory for the Dirac operator into the framework of smooth perturbations. Finally, we find the asymptotics for large times t of solutions u(x, t) of the time-dependent Dirac equation. Academic Press

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Section: Lemma 28 Suppose That Functions V and Amentioning

confidence: 97%

Scattering Theory for the Dirac Operator with a Long-Range Electromagnetic Potential

Gâtel

Yafaev

2001

Journal of Functional Analysis

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“…We also mention that Weyl's theorem gives that the essential spectrum of H is (−∞, −m] ∪ [+m, +∞), and the work of Berthier and Georgescu [5,Theorems 6 and A] gives that there is no embedded eigenvalue. Hence the thresholds ±m are the only points of the continuous spectrum which can be associated with a wave of zero velocity.…”

Section: Time Decay Estimatesmentioning

confidence: 99%

On the Asymptotic Stability of Small Nonlinear Dirac Standing Waves in a Resonant Case

Boussaïd¹

2008

SIAM J. Math. Anal.

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Abstract. We study the behavior of perturbations of small nonlinear Dirac standing waves. We assume that the linear Dirac operator of reference H = Dm + V has only two double eigenvalues and that degeneracies are due to a symmetry of H (theorem of Kramers). In this case, we can build a small four-dimensional manifold of stationary solutions tangent to the first eigenspace of H. Then we assume that a resonance condition holds, and we build a center manifold of real codimension 8 around each stationary solution. Inside this center manifold any H s perturbation of stationary solutions, with s > 2, stabilizes towards a standing wave. We also build center-stable and center-unstable manifolds, each one of real codimension 4. Inside each of these manifolds, we obtain stabilization towards the center manifold in one direction of time, while in the other, we have instability. Eventually, outside all of these manifolds, we have instability in the two directions of time. For localized perturbations inside the center manifold, we obtain a nonlinear scattering result. Introduction. We study the asymptotic stability of stationary solutions of a time-dependent nonlinear Dirac equation.A localized stationary solution of a given time-dependent equation represents a bound state of a particle. Like Rañada [39], we call it a particle-like solution (PLS). Many works have been devoted to the proof of the existence of such solutions for a wide variety of equations. Although their stability is a crucial problem (in particular, in a numerical computation or experiment), less attention has been devoted to this issue.In this paper, we deal with the problem of stability of small PLSs of the following nonlinear Dirac equation:where ∇F is the gradient of F : C 4 → R for the standard scalar product of R 8 . Here, D m is the usual Dirac operator (see Thaller [48]where m ∈ R * + , α = (α 1 , α 2 , α 3 ), and β are C 4 Hermitian matrices satisfying whereIn (0.1), V is the external potential field, and F : C 4 → R is a nonlinearity with the following gauge invariance:Some additional assumptions on F and V will be made in what follows. Nonlinearity with no potential arises in some Dirac models introduced by physicists to model either extended particles with self-interaction or particles in space-time with geometrical structure. In the latter case, physicists have shown that a relativistic theory sometimes imposes a fourth order nonlinear potential (i.e., a cubic nonlinearity) such as the square of a quadratic form on C 4 ; see Rañada [39] and the references therein. We added a potential with special features to ensure the existence of small stationary solutions, whose existence and stability are easier to study.Stationary solutions (PLSs) of (0.1) take the form ψ(t, x) = e −iEt φ(x), where φ satisfiesWe show that there exists a manifold of small solutions to (0.3) tangent to the first eigenspace of D m + V (see Proposition 1.1 below).Concerning the asymptotic stability in the Schrödinger equation, the question has been solved in several cases. Fo...

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“…where the ε i 's are eigenvalues of finite multiplicity in (−mc 2 , mc 2 ), which can only accumulate at the thresholds −mc 2 or mc 2 (see [18,19]). The min-max formulas presented in this section furnish a useful variational characterization of the ε i 's and of the associated eigenfunctions:…”

Section: Compared With Nonrelativistic Theories In Which the Schrödinmentioning

confidence: 99%

“…Note that one expects that for a reasonable potential there are no eigenvalues embedded in the essential spectrum. Very general conditions on V that ensure nonexistence of embedded eigenvalues have been given by [18,19]. Finally note that in this section c is kept variable.…”

Section: Linear Dirac Equations For An Electron In An External Fieldmentioning

confidence: 99%

Variational methods in relativistic quantum mechanics: new approach to the computation of Dirac eigenvalues

Dolbeault

Esteban

Séré

2000

Lecture Notes in Chemistry

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Abstract. This review is devoted to the study of stationary solutions of linear and nonlinear equations from relativistic quantum mechanics, involving the Dirac operator. The solutions are found as critical points of an energy functional. 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 us 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 KleinGordon-Dirac equations.The second part is devoted to the presentation of min-max principles allowing us to characterize and compute the eigenvalues of linear Dirac operators with an external potential in the gap of their essential spectrum. Many consequences of these min-max characterizations are presented, among them are new kinds 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 Quantum Electrodynamics in which the behavior of the vacuum is taken into account, 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.

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On the point spectrum of Dirac operators

Cited by 60 publications

References 4 publications

Scattering Theory for the Dirac Operator with a Long-Range Electromagnetic Potential

Scattering Theory for the Dirac Operator with a Long-Range Electromagnetic Potential

On the Asymptotic Stability of Small Nonlinear Dirac Standing Waves in a Resonant Case

Variational methods in relativistic quantum mechanics: new approach to the computation of Dirac eigenvalues

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