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Propagation estimates for \(N\)-body Stark Hamiltonians. (English) Zbl 0843.35087

The author establishes some propagation estimates for a system of \(N\) particles moving in a given constant electric field that interact with one another through pair potentials (\(N\)-body Stark Hamiltonians). One of the estimates implies that the particles asymptotically concentrate in any conical neighborhood of the electric field, and this fact has played an important role for the proof of the asymptotic completeness for long-range \(N\)-body Stark Hamiltonians. Also, the estimates can be used to prove the existence of the time-delay operator for two-body Stark Hamiltonians.
Reviewer: L.Vazquez (Madrid)

MSC:

35Q40 PDEs in connection with quantum mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35F10 Initial value problems for linear first-order PDEs
81V70 Many-body theory; quantum Hall effect
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References:

[1] T. Adachi , Long-range scattering for three-body Stark Hamiltonians , J. Math. Phys. , Vol. 35 , 1994 , pp. 5547 - 5571 . MR 1299904 | Zbl 0819.35106 · Zbl 0819.35106 · doi:10.1063/1.530764
[2] T. Adachi and H. Tamura , Asymptotic completeness for long-range many-particle systems with Stark Effect , to appear in J. Math. Sci. , The Univ. of Tokyo . MR 1348023 | Zbl 0843.35086 · Zbl 0843.35086
[3] T. Adachi and H. Tamura , Asymptotic completeness for long-range many-particle systems with Stark Effect, II , to appear in Commun. Math Phys. Article | MR 1370080 | Zbl 0849.35112 · Zbl 0849.35112 · doi:10.1007/BF02101527
[4] J. Dereziński , Asymptotic completeness of long-range N-body quantum systems , Ann. of Math. , Vol. 138 , 1993 , pp. 427 - 476 . MR 1240577 | Zbl 0844.47005 · Zbl 0844.47005 · doi:10.2307/2946615
[5] C. Gérard , Sharp propagation estimates for N-particle systems , Duke Math. J. , Vol. 67 , 1992 , pp. 483 - 515 . Article | MR 1181310 | Zbl 0760.35049 · Zbl 0760.35049 · doi:10.1215/S0012-7094-92-06719-6
[6] G.M. Graf , Asymptotic completeness for N-body short-range quantum systems: a new proof , Commun. Math. Phys. , Vol. 132 , 1990 , pp. 73 - 101 . Article | MR 1069201 | Zbl 0726.35096 · Zbl 0726.35096 · doi:10.1007/BF02278000
[7] G.M. Graf , A Remark on long-range Stark scattering , Helv. Phys. Acta. , Vol. 64 , 1991 , pp. 1167 - 1174 . MR 1149436
[8] B. Helffer and J. Sjöstrand , Equation de Schrödinger avec champ magnétique et équation de Harper , Lecture Notes in Physics 345 , Springer-Verlag , 1989 , pp. 118 - 197 . MR 1037319 | Zbl 0699.35189 · Zbl 0699.35189
[9] I.W. Herbst , J.S. Møller and E. Skibsted , Spectral analysis of N-body Stark Hamiltonians , Preprint, 1994 . · Zbl 0846.35095
[10] I.W. Herbst , J.S. Møller and E. Skibsted , Asymptotic completeness for N-body Stark Hamiltonians , to appear in Commun. Math Phys. Article | MR 1370079 | Zbl 0846.35096 · Zbl 0846.35096 · doi:10.1007/BF02101526
[11] A. Jensen and T. Ozawa , Existence and non-existence results for wave operators for perturbations of the Laplacian , Rev. Math. Phys. , Vol. 5 , 1993 , pp. 601 - 629 . MR 1240736 | Zbl 0831.35145 · Zbl 0831.35145 · doi:10.1142/S0129055X93000188
[12] A. Jensen and K. Yajima , On the long-range scattering for Stark Hamiltonians , J. Reine Angew. Math. , Vol. 420 , 1991 , pp. 179 - 193 . Article | MR 1124570 | Zbl 0736.35077 · Zbl 0736.35077
[13] D. Robert and X.P. Wang , Existence of time-delay operators for Stark Hamiltonians , Commun. Partial Differ. Eqs. , Vol. 14 , 1989 , pp. 63 - 98 . MR 973270 | Zbl 0677.35080 · Zbl 0677.35080 · doi:10.1080/03605308908820591
[14] I.M. Sigal and A. Soffer , The N-particle scattering problem: asymptotic completeness for short-range systems , Ann. of Math. , Vol. 125 , 1987 , pp. 35 - 108 . MR 898052 | Zbl 0646.47009 · Zbl 0646.47009 · doi:10.2307/1971345
[15] E. Skibsted , Propagation estimates for N-body Schroedinger operators , Commun. Math. Phys. , Vol. 142 , 1991 , pp. 67 - 98 . Article | MR 1137775 | Zbl 0760.35035 · Zbl 0760.35035 · doi:10.1007/BF02099172
[16] D. White , The Stark effect and long-range scattering in two Hilbert spaces , Indiana Univ. Math. J. , Vol. 39 , 1990 , pp. 517 - 546 . MR 1089052 | Zbl 0695.35144 · Zbl 0695.35144 · doi:10.1512/iumj.1990.39.39029
[17] D. White , Modified wave operators and Stark Hamiltonians , Duke Math. J. , Vol. 68 , 1992 , pp. 83 - 100 . Article | MR 1185819 | Zbl 0766.35033 · Zbl 0766.35033 · doi:10.1215/S0012-7094-92-06804-9
[18] L. Zielinski , A Proof of asymptotic completeness for N-body Schrödinger operators , Comm. Partial Differ. Eqs. , Vol. 19 , 1994 , pp. 455 - 522 . MR 1265807 | Zbl 0812.35118 · Zbl 0812.35118 · doi:10.1080/03605309408821024
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