no. 6
Propagation estimates for Dirac operators and application to scattering theory
Annales de l'Institut Fourier, Volume 54 (2004) no. 6, pp. 2021-2083.

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.

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.

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 
@article{AIF_2004__54_6_2021_0,
     author = {Daud\'e, Thierry},
     title = {Propagation estimates for {Dirac} operators and application to scattering theory},
     journal = {Annales de l'Institut Fourier},
     pages = {2021--2083},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {54},
     number = {6},
     year = {2004},
     doi = {10.5802/aif.2074},
     mrnumber = {2134232},
     zbl = {1080.35101},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2074/}
}
TY  - JOUR
TI  - Propagation estimates for Dirac operators and application to scattering theory
JO  - Annales de l'Institut Fourier
PY  - 2004
DA  - 2004///
SP  - 2021
EP  - 2083
VL  - 54
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2074/
UR  - https://www.ams.org/mathscinet-getitem?mr=2134232
UR  - https://zbmath.org/?q=an%3A1080.35101
UR  - https://doi.org/10.5802/aif.2074
DO  - 10.5802/aif.2074
LA  - en
ID  - AIF_2004__54_6_2021_0
ER  - 
%0 Journal Article
%T Propagation estimates for Dirac operators and application to scattering theory
%J Annales de l'Institut Fourier
%D 2004
%P 2021-2083
%V 54
%N 6
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2074
%R 10.5802/aif.2074
%G en
%F AIF_2004__54_6_2021_0
Daudé, Thierry. Propagation estimates for Dirac operators and application to scattering theory. Annales de l'Institut Fourier, Volume 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: 1388037 | Zbl: 0962.47500

[2] W. Amrein; A. Boutet de Monvel-Berthier; V. Georgescu On Mourre's approach of spectral theory, Helv. Phys. Acta, Tome 62 (1989), pp. 1-20 | MR: 991005 | Zbl: 0791.47026

[3] E. Baslev; B. Helffer Limiting absorption principle and resonances for the Dirac operator, Adv. in Appl. Math, Tome 13 (1992), pp. 186-215 | MR: 1162140 | Zbl: 0756.35062

[4] A. Berthier; V. Georgescu On the point spectrum of Dirac operators, J. Func. Anal, Tome 71 (1987), pp. 309-338 | MR: 880983 | Zbl: 0655.47043

[5] A. Boutet de Monvel-Berthier; D. Manda; R. Purice Limiting absorption principle for the Dirac operator, Ann. Inst. Henri Poincaré, Physique Théorique, Tome 58 (1993) no. 4, pp. 413-431 | EuDML: 76612 | Numdam | MR: 1241704 | Zbl: 0789.35134

[6] E.B. Davies Spectral Theory and Differential Operators, Cambridge studies in advanced mathematics, Tome 2 (1995) | MR: 1349825 | Zbl: 0893.47004

[7] J. Derezi#x0144;ski Asymptotic completeness for N-particle long-range quantum sytems, Ann. of Math, Tome 138 (1993), pp. 427-476 | MR: 1240577 | Zbl: 0844.47005

[8] J. Derezi#x0144;ski; C. Gérard Scattering Theory of Classical and Quantum N-Particle Systems, Springer-Verlag, 1997 | MR: 1459161 | Zbl: 0899.47007

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

[10] V. Enss Asymptotic completeness for quantum-mechanical potential scattering, I: Short range potentials., Comm. Math. Phys, Tome 61 (1978), pp. 285-291 | MR: 523013 | Zbl: 0389.47005

[11] V. Enss Asymptotic completeness for quantum-mechanical potential scattering, II: Singular and Long range potentials, Ann. Phys, Tome 119 (1979), pp. 117-132 | MR: 535624 | Zbl: 0408.47009

[12] V. Enss; B. Thaller Asymptotic observables and Coulomb scattering for the Dirac equation, Ann. Inst. Henri Poincaré. Physique Théorique, Tome 45 (1986), pp. 147-171 | Numdam | MR: 866913 | Zbl: 0615.47008

[13] Y. Gâtel; D.R. Yafaev Scattering theory for the Dirac operator with a long-range electromagnetic potential, J. Func. Anal, Tome 184 (2001) no. 1, pp. 136-176 | MR: 1846785 | Zbl: 0996.35056

[14] V. Georgescu; C. Gérard On the virial theorem in Quantum Mechanics, Comm. Math. Phys, Tome 208 (1999), pp. 275-281 | MR: 1729087 | Zbl: 0961.81009

[15] V. Georgescu; M. Mântoiu On the spectral theory of singular Dirac type hamiltonians, J. Operator Theory, Tome 46 (2001) no. 2, pp. 289-321 | MR: 1870409 | Zbl: 0993.35070

[16] C. Gérard; I. Laba Multiparticle quantum scattering in constant magnetic fields, Mathematical surveys and monographs, Tome 90, American Mathematical Society, 2002 | MR: 1871447 | Zbl: 1044.81123

[17] C. Gérard; F. Nier Scattering theory for the perturbation of periodic Schrödinger operators, J. Math. Kyoto Univ, Tome 38 (1998), pp. 595-634 | MR: 1669979 | Zbl: 0934.35111

[18] G.M. Graf Asymptotic completeness for N-body short range quantum systems: A new proof, Comm. Math. Phys, Tome 132 (1990), pp. 73-101 | MR: 1069201 | Zbl: 0726.35096

[19] D. Häfner Sur la théorie de la diffusion pour l'équation de Klein-Gordon dans la métrique de Kerr, Dissertationes Mathematicae, Tome 421 (2003) | MR: 2031494 | Zbl: 02120135

[20] D. Häfner; J.-P. Nicolas Scattering of massless Dirac fields by a Kerr black hole, Rev. Math. Phys, Tome 16 (2004) no. 1, pp. 29-123 | MR: 2047861 | Zbl: 1064.83036

[21] B. Helffer; J. Sjöstrand Equation de Schrödinger avec champ magnétique et équation de Harper, Lecture Notes in Physics, Tome 345 (1989), pp. 118-197 | MR: 1037319 | Zbl: 0699.35189

[22] W. Hunziker; I.M. Sigal; A. Soffer Minimal escape velocities, Comm. Partial Diff. Equ, Tome 24 (1999) no. 11-12, pp. 2279-2295 | MR: 1720738 | Zbl: 0944.35014

[23] A. Iftimovici; M. Mântoiu Limiting Absorption Principle at Critical Values for the Dirac Operator, Lett. Math. Phys, Tome 49 (1999) no. 3, pp. 235-243 | MR: 1743451 | Zbl: 0957.47029

[24] F. Melnyk Scattering on Reissner-Nordström metric for massive charged spin 1 2 fields, Ann. Henri Poincaré, Tome 4 (2003) no. 5, pp. 813-846 | MR: 2016993 | Zbl: 02038428

[25] E. Mourre Absence of singular continuous spectrum for certain self-adjoint operators, Comm. Math. Phys, Tome 78 (1981), pp. 391-408 | MR: 603501 | Zbl: 0489.47010

[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, Tome 50 (1986) no. 1-4, pp. 103-130 | MR: 989018 | Zbl: 0688.35070

[27] J.-P. Nicolas Scattering of linear Dirac fields by a pherically symmetric Black-Hole, Ann. Inst. Henri Poincaré. Physique Théorique, Tome 62 (1995) no. 2, pp. 145-179 | Numdam | MR: 1317184 | Zbl: 0826.53072

[28] M. Reed; B. Simon Methods of modern mathematical physics. I, Academic Press, 1972 | MR: 751959 | Zbl: 0401.47001

[28] M. Reed; B. Simon Methods of modern mathematical physics. II, Academic Press, 1975 | MR: 751959 | Zbl: 0401.47001

[28] M. Reed; B. Simon Methods of modern mathematical physics. III, Academic Press, 1979 | MR: 529429 | Zbl: 0405.47007

[28] M. Reed; B. Simon Methods of modern mathematical physics. IV, Academic Press, 1978 | MR: 751959 | Zbl: 0401.47001

[29] D. Ruelle A remark on bound states in potential scattering theory, Nuovo Cimento, A, Tome 61 (1969) | MR: 246603

[30] I.M. Sigal; A. Soffer The N-particle scattering problem: asymptotic completeness for short-range quantum systems, Ann. of Math, Tome 125 (1987), pp. 35-108 | MR: 898052 | Zbl: 0646.47009

[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: 1219537 | Zbl: 0765.47023

Cited by Sources: