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Propagation of singularities in many-body scattering. (English) Zbl 0987.81114

Author’s summary: We describe the propagation of singularities of tempered distributional solutions \(u\in {\mathcal S}'\) of \((H-\lambda) u=0\), \(\lambda>0\), where \(H\) is a many-body Hamiltonian \(H=\Delta +V\), \(\Delta\geq 0\), \(V=\sum_a V_a\), under the assumption that no subsystem has a bound state and that the two-body interactions \(V_a\) are real-valued polyhomogeneous symbols of order \(-1\) (e.g. Coulomb-type with the singularity at the origin removed). Here the term ‘singularity’ provides a microlocal description of the lack of decay at infinity. We use this result to prove that the wave front relation of the free-to-free \(S\)-matrix (which, under our assumptions, is all of the \(S\)-matrix) is given by the broken geodesic flow, broken at the ‘singular directions’, on \(\mathbb{S}^{n-1}\) at time \(\pi\). We also present a natural geometric generalization to asymptotically Euclidean spaces.

MSC:

81U10 \(n\)-body potential quantum scattering theory
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[1] Bommier A. , Propriétés de la matrice de diffusion , Ann. Inst. Henri Poincaré 59 ( 1993 ) 237 - 267 . Numdam | MR 1276326 | Zbl 0804.35111 · Zbl 0804.35111
[2] Dereziński J. , Asymptotic completeness of long-range N -body quantum systems , Ann. Math. 138 ( 1993 ) 427 - 476 . MR 1240577 | Zbl 0844.47005 · Zbl 0844.47005 · doi:10.2307/2946615
[3] Dereziński J. , Gérard C. , Scattering Theory of Classical Quantum N -particle Systems , Springer , 1997 . MR 1459161 | Zbl 0899.47007 · Zbl 0899.47007
[4] Froese R.G. , Herbst I. , Exponential bounds , Comm. Math. Phys. 87 ( 1982 ) 429 - 447 . Article | MR 682117 | Zbl 0509.35061 · Zbl 0509.35061 · doi:10.1007/BF01206033
[5] Froese R.G. , Herbst I. , A new proof of the Mourre estimate , Duke Math. J. 49 ( 1982 ) 1075 - 1085 . Article | MR 683011 | Zbl 0514.35025 · Zbl 0514.35025 · doi:10.1215/S0012-7094-82-04947-X
[6] Gérard C. , Isozaki H. , Skibsted E. , Commutator algebra resolvent estimates , in: Advanced Studies in Pure Mathematics, Vol. 23 , 1994 , pp. 69 - 82 . MR 1275395 · Zbl 0814.35086
[7] Gérard C. , Isozaki H. , Skibsted E. , N -body resolvent estimates , J. Math. Soc. Japan 48 ( 1996 ) 135 - 160 . Article | MR 1361552 | Zbl 0851.35101 · Zbl 0851.35101 · doi:10.2969/jmsj/04810135
[8] Graf G.M. , Asymptotic completeness for N -body short range systems , Comm. Math. Phys. 132 ( 1990 ) 73 - 101 . Article | MR 1069201 | Zbl 0726.35096 · Zbl 0726.35096 · doi:10.1007/BF02278000
[9] Hassell A. , Vasy A. , Symbolic functional calculus , J. Funct. Anal. 173 ( 2000 ) 257 - 283 . MR 1760615 | Zbl 0960.58025 · Zbl 0960.58025 · doi:10.1006/jfan.2000.3569
[10] Helffer B. , Sjöstrand J. , Équation de Schrödinger avec champ magnétique et équation de Harper. Zbl 0641.35015 · Zbl 0641.35015
[11] Hörmander L. , Fourier integral operators , Acta Mathematica 127 ( 1971 ) 79 - 183 . MR 388463 | Zbl 0212.46601 · Zbl 0212.46601 · doi:10.1007/BF02392052
[12] Hörmander L. , The Analysis of Linear Partial Differential Operators, Vols. 1-4 , Springer-Verlag , 1983 .
[13] Ikawa M. (Ed.), Spectral and Scattering Theory , Marcel Dekker , 1994 . MR 1291635 | Zbl 0798.00016 · Zbl 0798.00016
[14] Isozaki H. , Structures of S-matrices for three body Schrödinger operators , Comm. Math. Phys. 146 ( 1992 ) 241 - 258 . Article | MR 1165182 | Zbl 0748.35026 · Zbl 0748.35026 · doi:10.1007/BF02102627
[15] Isozaki H. , A generalization of the radiation condition of Sommerfeld for N -body Schrödinger operators , Duke Math. J. 74 ( 1994 ) 557 - 584 . Article | MR 1272984 | Zbl 0811.35107 · Zbl 0811.35107 · doi:10.1215/S0012-7094-94-07420-6
[16] Isozaki H. , A uniqueness theorem for the N -body Schrödinger equation and its applications , in: Ikawa M. (Ed.), Spectrial and Scattering Theory , Marcel Dekker , 1994 . MR 1291639 | Zbl 0813.35068 · Zbl 0813.35068
[17] Jensen A. , Propagation estimates for Schrödinger-type operators , Trans. Amer. Math. Soc. 291 ( 1 ) ( 1985 ) 129 - 144 . MR 797050 | Zbl 0577.35089 · Zbl 0577.35089 · doi:10.2307/1999899
[18] Lebeau G. , Propagation des ondes dans les variétés à coins , Ann. Scient. Éc. Norm. Sup. 30 ( 1997 ) 429 - 497 . Numdam | MR 1456242 | Zbl 0891.35072 · Zbl 0891.35072 · doi:10.1016/S0012-9593(97)89928-4
[19] Mazzeo R. , Melrose R.B. , Pseudodifferential operators on manifolds with fibred boundaries , Asian J. Math. 2 ( 1998 ). MR 1734130 | Zbl 01531011 · Zbl 1125.58304
[20] Melrose R.B. , Differential analysis on manifolds with corners, In preparation. · Zbl 0754.58035
[21] Melrose R.B. , Calculus of conormal distributions on manifolds with corners , Internat. Math. Res. Notices 3 ( 1992 ) 51 - 61 . MR 1154213 | Zbl 0754.58035 · Zbl 0754.58035 · doi:10.1155/S1073792892000060
[22] Melrose R.B. , Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces , in: Ikawa M. (Ed.), Spectrial and Scattering Theory , Marcel Dekker , 1994 . MR 1291640 | Zbl 0837.35107 · Zbl 0837.35107
[23] Melrose R.B. , Sjöstrand J. , Singularities of boundary value problems. I , Comm. Pure Appl. Math. 31 ( 1978 ) 593 - 617 . MR 492794 | Zbl 0368.35020 · Zbl 0368.35020 · doi:10.1002/cpa.3160310504
[24] Melrose R.B. , Sjöstrand J. , Singularities of boundary value problems. II , Comm. Pure Appl. Math. 35 ( 1982 ) 129 - 168 . MR 644020 | Zbl 0546.35083 · Zbl 0546.35083 · doi:10.1002/cpa.3160350202
[25] Melrose R.B. , Zworski M. , Scattering metrics , Inventiones Math. 124 ( 1996 ) 389 - 436 . MR 1369423 | Zbl 0855.58058 · Zbl 0855.58058 · doi:10.1007/s002220050058
[26] Mourre E. , Absence of singular continuous spectrum of certain self-adjoint operators , Comm. Math. Phys. 78 ( 1981 ) 391 - 408 . Article | MR 603501 | Zbl 0489.47010 · Zbl 0489.47010 · doi:10.1007/BF01942331
[27] Perry P. , Sigal I.M. , Simon B. , Spectral analysis of N -body Schrödinger operators , Ann. Math. 114 ( 1981 ) 519 - 567 . MR 634428 | Zbl 0477.35069 · Zbl 0477.35069 · doi:10.2307/1971301
[28] Sigal I.M. , Soffer A. , N -particle scattering problem , Ann. Math. 125 ( 1987 ) 35 - 108 . MR 898052 | Zbl 0646.47009 · Zbl 0646.47009 · doi:10.2307/1971345
[29] Sigal I.M. , Soffer A. , Long-range many-body scattering , Inventiones Math. 99 ( 1990 ) 115 - 143 . MR 1029392 | Zbl 0702.35197 · Zbl 0702.35197 · doi:10.1007/BF01234413
[30] Sigal I.M. , Soffer A. , Asymptotic completeness of N \leq 4-particle systems with the Coulomb-type interactions , Duke Math. J. 71 ( 1993 ) 243 - 298 . Article | Zbl 0853.70010 · Zbl 0853.70010 · doi:10.1215/S0012-7094-93-07110-4
[31] Sigal I.M. , Soffer A. , Asymptotic completeness of N -particle long-range scattering , J. Amer. Math. Soc. 7 ( 1994 ) 307 - 334 . MR 1233895 | Zbl 0811.35091 · Zbl 0811.35091 · doi:10.2307/2152761
[32] Sjöstrand J. , Propagation of analytic singularities for second order Dirichlet problems , Comm. PDE 5 ( 1980 ) 41 - 94 . MR 556454 | Zbl 0458.35026 · Zbl 0458.35026 · doi:10.1080/03605308008820133
[33] Skibsted E. , Smoothness of N -body scattering amplitudes , Rev. Math. Phys. 4 ( 1992 ) 619 - 658 . MR 1197552 | Zbl 0781.35047 · Zbl 0781.35047 · doi:10.1142/S0129055X92000248
[34] Taylor M. , Grazing rays , Comm. Pure Appl. Math. 29 ( 1976 ) 1 - 38 . MR 397175 | Zbl 0318.35009 · Zbl 0318.35009 · doi:10.1002/cpa.3160290102
[35] Vasy A. , Asymptotic behavior of generalized eigenfunctions in N -body scattering , J. Funct. Anal. 148 ( 1997 ) 170 - 184 . MR 1461498 | Zbl 0884.35110 · Zbl 0884.35110 · doi:10.1006/jfan.1996.3053
[36] Vasy A. , Propagation of singularities in three-body scattering , Ph.D. Thesis, Massachusetts Institute of Technology , 1997 . · Zbl 0884.35110
[37] Vasy A. , Structure of the resolvent for three-body potentials , Duke Math. J. 90 ( 1997 ) 379 - 434 . Article | MR 1484859 | Zbl 0891.35111 · Zbl 0891.35111 · doi:10.1215/S0012-7094-97-09010-4
[38] Vasy A. , Geometric scattering theory for long-range potentials , Int. Math. Res. Notices 1998 ( 1998 ) 285 - 315 . MR 1616722 | Zbl 0922.58085 · Zbl 0922.58085 · doi:10.1155/S1073792898000208
[39] Vasy A. , Scattering matrices in many-body scattering , Comm. Math. Phys. 200 ( 1999 ) 105 - 124 . MR 1671912 | Zbl 0929.47044 · Zbl 0929.47044 · doi:10.1007/s002200050524
[40] Vasy A. , Propagation of singularities in three-body scattering , Astérisque 262 ( 2000 ). MR 1744795 | Zbl 0941.35001 · Zbl 0941.35001
[41] Wang X.P. , Microlocal estimates for N -body Schrödinger operators , J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 40 ( 1993 ) 337 - 385 . MR 1255046 | Zbl 0810.35073 · Zbl 0810.35073
[42] Yafaev D. , Radiation conditions , Comm. Math. Phys. 154 ( 1993 ) 523 - 554 . Article | MR 1224090 | Zbl 0781.35048 · Zbl 0781.35048 · doi:10.1007/BF02102107
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