Scattering theory for 3-particle systems in constant magnetic fields: dispersive case
Annales de l'Institut Fourier, Tome 46 (1996) no. 3, pp. 801-876.

Nous développons la théorie de la diffusion pour des systèmes quantiques de trois particules chargées en présence d’un champ magnétique constant. Nous généralisons nos travaux précèdents en ne faisant pas d’autres hypothèses sur les charges des sous systèmes. La difficulté principale est dans l’analyse des canaux de diffusion correspondant au mouvement des états liés des sous systèmes neutres transversalement au champ magnétique. L’énergie cinétique effective de ce mouvement est donnée par certains hamiltoniens dispersifs. Sous des hypothèses convenables sur la régularité des valeurs propres des hamiltoniens réduits, nous obtenons une estimation de Mourre ainsi que la complétude asymptotique pour des interactions à courte portée et de type de Coulomb.

We develop a scattering theory for quantum systems of three charged particles in a constant magnetic field. For such systems, we generalize our earlier results in that we make no additional assumptions on the electric charges of subsystems. The main difficulty is the analysis of the scattering channels corresponding to the motion of the bound states of the neutral subsystems in the directions transversal to the field. The effective kinetic energy of this motion is given by certain dispersive Hamiltonians; therefore we refer to this case as dispersive. Under suitable assumptions on the regularity of the eigenvalues of the reduced Hamiltonians, we obtain the Mourre estimate for general long-range systems, and asymptotic completeness for short-range and Coulomb systems.

@article{AIF_1996__46_3_801_0,
     author = {G\'erard, Christian and {\L}aba, Izabella},
     title = {Scattering theory for 3-particle systems in constant magnetic fields: dispersive case},
     journal = {Annales de l'Institut Fourier},
     pages = {801--876},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {46},
     number = {3},
     year = {1996},
     doi = {10.5802/aif.1532},
     zbl = {0853.35098},
     mrnumber = {97j:81377},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1532/}
}
TY  - JOUR
AU  - Gérard, Christian
AU  - Łaba, Izabella
TI  - Scattering theory for 3-particle systems in constant magnetic fields: dispersive case
JO  - Annales de l'Institut Fourier
PY  - 1996
SP  - 801
EP  - 876
VL  - 46
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1532/
DO  - 10.5802/aif.1532
LA  - en
ID  - AIF_1996__46_3_801_0
ER  - 
%0 Journal Article
%A Gérard, Christian
%A Łaba, Izabella
%T Scattering theory for 3-particle systems in constant magnetic fields: dispersive case
%J Annales de l'Institut Fourier
%D 1996
%P 801-876
%V 46
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1532/
%R 10.5802/aif.1532
%G en
%F AIF_1996__46_3_801_0
Gérard, Christian; Łaba, Izabella. Scattering theory for 3-particle systems in constant magnetic fields: dispersive case. Annales de l'Institut Fourier, Tome 46 (1996) no. 3, pp. 801-876. doi : 10.5802/aif.1532. https://aif.centre-mersenne.org/articles/10.5802/aif.1532/

[Ag] S. Agmon, Lectures on exponential decay of solutions of second order elliptic equations, Princeton University Press, Princeton, 1982. | Zbl

[AHS1] J. Avron, I. Herbst, B. Simon, Schrödinger operators with magnetic fields I : General interactions, Duke Math. J., 45 (1978), 847-884. | MR | Zbl

[AHS2] J. Avron, I. Herbst, B. Simon, Schrödinger operators with magnetic fields II : Separation of the center of mass in homogeneous magnetic fields, Ann. Phys., 114 (1978), 431-451. | MR | Zbl

[BG] A. Boutet De Monvel, I.V. Georgescu, Spectral and scattering theory by the conjugate operator method, Algebra and Analysis, Vol 4 (1992), 73-116. | Zbl

[BP] A. Boutet De Monvel, R. Purice, Limiting absorption principle for Schrödinger Hamiltonians with magnetic fields, Comm. in P.D.E., 19 (1994), 89-117. | MR | Zbl

[CFKS] H.L. Cycon, R. Froese, W. Kirsch, B. Simon, Schrödinger operators with applications to Quantum Mechanics and Global Geometry, Texts and Monographs in Physics, Springer, 1987. | Zbl

[Do] J. Dollard, Asymptotic convergence and Coulomb interaction, J. Math. Phys., Vol 5 (1964), 729-738.

[De1] J. Dereziński, Asymptotic completeness for N-particle long -range quantum systems, Ann. Math., 138 (1993), 427-476. | MR | Zbl

[De2] J. Dereziński, The Mourre estimate for dispersive N-body Schrödinger operators, Trans. of AMS, 317 (1990), 773-798. | MR | Zbl

[DG] J. Dereziński, C. Gérard, Asymptotic completeness of N-particle systems, chapter 3, preprint Erwin Schrödinger Institute, 1993.

[E1] V. Enss, Asymptotic completeness for quantum-mechanical potential scattering, I. Short-range potentials, Comm. Math. Phys., 61 (1978), 285-291. | MR | Zbl

[E2] V. Enss, Quantum scattering theory for two- and three-body systems with potentials of short and long range, in : Schrödinger Operators, S.Graffi ed., Springer LN Math. 1218, Berlin, 1986.

[E3] V. Enss, Quantum scattering with long-range magnetic fields. Operator Theory, Adv. and Appl., 57 (1993), 61-70. | MR | Zbl

[FH] R. Froese, I. Herbst, A new proof of the Mourre estimate, Duke Math. J., 49 (1982), 1075-1085. | MR | Zbl

[G] C. Gérard, Sharp propagation estimates for N-particle systems, Duke Math. J., 67 (1992), 483-515. | MR | Zbl

[GL1] C. Gérard, I. Laba, Scattering theory for N-particle systems in constant magnetic fields, Duke Math. J., 76 (1994), 433-465. | MR | Zbl

[GL2] C. Gérard, I. Laba, Scattering theory for N-particle systems in constant magnetic fields, II. Long-range interactions, Comm. in PDE, 20 (1995), 1791-1830. | MR | Zbl

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

[HeSj] B. Helffer, J. Sjöstrand, Equation de Schrödinger avec champ magnétique et équation de Harper, Springer Lectures Notes in Physics n° 345 (1989), 118-197. | Zbl

[Hö] L. Hörmander, The analysis of linear partial differential operators, volume III Springer Verlag (1985). | Zbl

[HuSi] W. Hunziker, I.M. Sigal, The General Theory of N-Body Quantum Systems, in preparation. | Zbl

[IK] H. Isozaki, H. Kitada, Modified wave operators with time-dependent modifiers. J. Fac. Sci. Univ. Tokyo, Sec. 1A, 32 (1985), 77-104. | MR | Zbl

[I1] H. Iwashita, On the long-range scattering for one- and two-particle Schrödinger operators with constant magnetic fields, Preprint, Nagoya Institute of Technology, 1993.

[I2] H. Iwashita, Spectral theory for 3-particle quantum systems with constant magnetic fields, Preprint, Nagoya Institute of Technology, 1993.

[KY] H. Kitada, K. Yajima, A scattering theory for time-dependent long-range potentials, Duke Math. J., 49 (1982), 341-376 and 50 (1983), 1005-1016. | MR | Zbl

[L1] I. Laba, Long-range one-particle scattering in a homogeneous magnetic field, Duke Math. J., 70 (1993), 283-303. | MR | Zbl

[L2] I. Laba, Scattering for hydrogen-like systems in a constant magnetic field, Comm. in P.D.E., 20 (1995), 741-762. | MR | Zbl

[LT1] M. Loss, B. Thaller, Scattering of particles by long-range magnetic fields. Ann. Phys., 176 (1987), 159-180. | MR | Zbl

[LT2] M. Loss, B. Thaller, Short-range scattering in long-range magnetic fields : the relativistic case, J. Diff. Equ., 73 (1988), 225-236. | MR | Zbl

[M] Y. Meyer, Ondelettes et Opérateurs II, Hermann Ed., 1990. | Zbl

[Mo] E. Mourre, Absence of singular continuous spectrum for certain selfadjoint operators, Comm. in Math. Phys., 78 (1981), 519-567. | MR | Zbl

[Ni] F. Nicoleau, Théorie de la diffusion pour l'opérateur de Schrödinger en présence d'un champ magnétique, Thèse de Doctorat de l'Université de Nantes, 1991.

[NR] F. Nicoleau, D. Robert, Théorie de la diffusion quantique pour des perturbations à longue et courte portée du champ magnétique, Ann. Fac. Sci. de Toulouse, 12 (1991), 185-194. | Numdam | MR | Zbl

[PSS] P. Perry, I.M. Sigal, B. Simon, Spectral analysis of N-body Schrödinger operators, Ann. Math., 114 (1981), 519-567. | MR | Zbl

[RSII] M. Reed, B. Simon, Methods of modern mathematical physics, Vol II, Academic Press. | Zbl

[SS1] I.M. Sigal, A. Soffer, The N-particle scattering problem : asymptotic completeness for short-range quantum systems, Ann. Math., 125 (1987), 35-108. | MR | Zbl

[SS2] I.M. Sigal, A. Soffer, Long-range many-body scattering. Asymptotic clustering for Coulomb-type potentials, Invent. Math., 99 (1990), 115-143. | MR | Zbl

[SS3] I.M. Sigal, A. Soffer, Local decay and velocity bounds, preprint, Princeton University, 1988.

[SS4] I.M. Sigal, A. Soffer, Asymptotic completeness for four-body Coulomb systems, Duke Math. J., 71 (1993), 243-298. | MR | Zbl

[SS5] I.M. Sigal, A. Soffer, Asymptotic completeness of N particle long range scattering, J. AMS., 7 (1994), 307-334. | MR | Zbl

[S] B. Simon, Phase space analysis of simple scattering systems : extensions of some work of Enss, Duke Math. J., 46 (1979), 119-168. | MR | Zbl

[VZ] S.A. Vugalter, G.M. Zhislin, Localization of the essential spectrum of the energy operators of n-particle quantum systems in a magnetic field, Theor. and Math. Phys., 97(1) (1993), 1171-1185. | MR | Zbl

Cité par Sources :