no. 6
Propagation estimates for Dirac operators and application to scattering theory
[Estimations de propagation pour des opérateurs de Dirac et application à la théorie de la diffusion]
Daudé, Thierry1
1 Université Bordeaux I, Institut de Mathématiques Appliquées de Bordeaux, UMR CNRS 5466, 351 cours de la libération, 33405 Talence Cedex (France)
Annales de l'Institut Fourier, Tome 54 (2004) no. 6, pp. 2021-2083.

Dans cet article, nous prouvons plusieurs estimations de propagation pour une équation de Dirac massive en espace-temps plat. Ces estimations nous permettent de construire l'opérateur de vitesse asymptotique et de caractériser son spectre. En utilisant cette nouvelle information, nous obtenons des résultats complets de scattering. Précisèment, nous prouvons l'existence et la complétude asymptotique des opérateurs d'onde modifiés à la Dollard.

In this paper, we prove propagation estimates for a massive Dirac equation in flat spacetime. This allows us to construct the asymptotic velocity operator and to analyse its spectrum. Eventually, using this new information, we are able to obtain complete scattering results; that is to say we prove the existence and the asymptotic completeness of the Dollard modified wave operators.

DOI : 10.5802/aif.2074
Classification : 35P25, 35Q40, 35B40, 81U99
Keywords: Partial differential equations, spectral theory, scattering theory, Dirac's equation, propagation estimates, Mourre theory
Mot clés : Equations aux dérivées partielles, théorie spectrale, théorie de la diffusion, équation de Dirac, estimations de propagation, théorie de Mourre

Daudé, Thierry 1

1 Université Bordeaux I, Institut de Mathématiques Appliquées de Bordeaux, UMR CNRS 5466, 351 cours de la libération, 33405 Talence Cedex (France) 
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Daudé, Thierry. Propagation estimates for Dirac operators and application to scattering theory. Annales de l'Institut Fourier, Tome 54 (2004) no. 6, pp. 2021-2083. doi : 10.5802/aif.2074. https://aif.centre-mersenne.org/articles/10.5802/aif.2074/

[1] W. Amrein; A. Boutet de Monvel; V. Georgescu C 0 -groups, commutator methods and spectral theory of N-body hamiltonians, Birkhäuser Verlag, 1996 | MR | Zbl

[2] W. Amrein; A. Boutet de Monvel-Berthier; V. Georgescu On Mourre's approach of spectral theory, Helv. Phys. Acta, Volume 62 (1989), pp. 1-20 | MR | Zbl

[3] E. Baslev; B. Helffer Limiting absorption principle and resonances for the Dirac operator, Adv. in Appl. Math, Volume 13 (1992), pp. 186-215 | MR | Zbl

[4] A. Berthier; V. Georgescu On the point spectrum of Dirac operators, J. Func. Anal, Volume 71 (1987), pp. 309-338 | MR | Zbl

[5] A. Boutet de Monvel-Berthier; D. Manda; R. Purice Limiting absorption principle for the Dirac operator, Ann. Inst. Henri Poincaré, Physique Théorique, Volume 58 (1993) no. 4, pp. 413-431 | EuDML | Numdam | MR | Zbl

[6] E.B. Davies Spectral Theory and Differential Operators, Cambridge studies in advanced mathematics, Volume 2 (1995) | MR | Zbl

[7] J. Derezi#x0144;ski Asymptotic completeness for N-particle long-range quantum sytems, Ann. of Math, Volume 138 (1993), pp. 427-476 | MR | Zbl

[8] J. Derezi#x0144;ski; C. Gérard Scattering Theory of Classical and Quantum N-Particle Systems, Springer-Verlag, 1997 | MR | Zbl

[9] J. Dollard; G. Velo Asymptotic behaviour of a Dirac particle in a Coulomb field, II, Nuovo Cimento, Volume 45 (1966), pp. 801-812

[10] V. Enss Asymptotic completeness for quantum-mechanical potential scattering, I: Short range potentials., Comm. Math. Phys, Volume 61 (1978), pp. 285-291 | MR | Zbl

[11] V. Enss Asymptotic completeness for quantum-mechanical potential scattering, II: Singular and Long range potentials, Ann. Phys, Volume 119 (1979), pp. 117-132 | MR | Zbl

[12] V. Enss; B. Thaller Asymptotic observables and Coulomb scattering for the Dirac equation, Ann. Inst. Henri Poincaré. Physique Théorique, Volume 45 (1986), pp. 147-171 | Numdam | MR | Zbl

[13] Y. Gâtel; D.R. Yafaev Scattering theory for the Dirac operator with a long-range electromagnetic potential, J. Func. Anal, Volume 184 (2001) no. 1, pp. 136-176 | MR | Zbl

[14] V. Georgescu; C. Gérard On the virial theorem in Quantum Mechanics, Comm. Math. Phys, Volume 208 (1999), pp. 275-281 | MR | Zbl

[15] V. Georgescu; M. Mântoiu On the spectral theory of singular Dirac type hamiltonians, J. Operator Theory, Volume 46 (2001) no. 2, pp. 289-321 | MR | Zbl

[16] C. Gérard; I. Laba Multiparticle quantum scattering in constant magnetic fields, Mathematical surveys and monographs, 90, American Mathematical Society, 2002 | MR | Zbl

[17] C. Gérard; F. Nier Scattering theory for the perturbation of periodic Schrödinger operators, J. Math. Kyoto Univ, Volume 38 (1998), pp. 595-634 | MR | Zbl

[18] G.M. Graf Asymptotic completeness for N-body short range quantum systems: A new proof, Comm. Math. Phys, Volume 132 (1990), pp. 73-101 | MR | Zbl

[19] D. Häfner Sur la théorie de la diffusion pour l'équation de Klein-Gordon dans la métrique de Kerr, Dissertationes Mathematicae, Volume 421 (2003) | MR | Zbl

[20] D. Häfner; J.-P. Nicolas Scattering of massless Dirac fields by a Kerr black hole, Rev. Math. Phys, Volume 16 (2004) no. 1, pp. 29-123 | MR | Zbl

[21] B. Helffer; J. Sjöstrand Equation de Schrödinger avec champ magnétique et équation de Harper, Lecture Notes in Physics, Volume 345 (1989), pp. 118-197 | MR | Zbl

[22] W. Hunziker; I.M. Sigal; A. Soffer Minimal escape velocities, Comm. Partial Diff. Equ, Volume 24 (1999) no. 11-12, pp. 2279-2295 | MR | Zbl

[23] A. Iftimovici; M. Mântoiu Limiting Absorption Principle at Critical Values for the Dirac Operator, Lett. Math. Phys, Volume 49 (1999) no. 3, pp. 235-243 | MR | Zbl

[24] F. Melnyk Scattering on Reissner-Nordström metric for massive charged spin 1 2 fields, Ann. Henri Poincaré, Volume 4 (2003) no. 5, pp. 813-846 | MR | Zbl

[25] E. Mourre Absence of singular continuous spectrum for certain self-adjoint operators, Comm. Math. Phys, Volume 78 (1981), pp. 391-408 | MR | Zbl

[26] PL. Muthuramalingam; K.B. Sinha Existence and completeness of wave operators for the Dirac operator in an electro-magnetic field with long range potentials, J. Indian Math. Soc, Volume 50 (1986) no. 1-4, pp. 103-130 | MR | Zbl

[27] J.-P. Nicolas Scattering of linear Dirac fields by a pherically symmetric Black-Hole, Ann. Inst. Henri Poincaré. Physique Théorique, Volume 62 (1995) no. 2, pp. 145-179 | Numdam | MR | Zbl

[28] M. Reed; B. Simon Methods of modern mathematical physics. I, Academic Press, 1972 | MR | Zbl

[28] M. Reed; B. Simon Methods of modern mathematical physics. II, Academic Press, 1975 | MR | Zbl

[28] M. Reed; B. Simon Methods of modern mathematical physics. III, Academic Press, 1979 | MR | Zbl

[28] M. Reed; B. Simon Methods of modern mathematical physics. IV, Academic Press, 1978 | MR | Zbl

[29] D. Ruelle A remark on bound states in potential scattering theory, Nuovo Cimento, A, Volume 61 (1969) | MR

[30] I.M. Sigal; A. Soffer The N-particle scattering problem: asymptotic completeness for short-range quantum systems, Ann. of Math, Volume 125 (1987), pp. 35-108 | MR | Zbl

[31] I.M. Sigal; A. Soffer Local decay and velocity bounds (1988) (Preprint, Princeton University)

[32] B. Thaller The Dirac Equation, Texts and monographs in Physics, Springer-Verlag, 1992 | MR | Zbl

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