N-body resolvent estimates

Christian GÉRARD; Hiroshi ISOZAKI; Erik SKIBSTED

doi:10.2969/jmsj/04810135

Christian GÉRARD, Hiroshi ISOZAKI, Erik SKIBSTED

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JOURNAL FREE ACCESS

1996 Volume 48 Issue 1 Pages 135-160

DOI https://doi.org/10.2969/jmsj/04810135

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Published: 1996 Received: April 14, 1994 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : B) A. Bommier, Propriétés de la matrice de diffusion, 2-auras-2-amas, pour les problèmes à N corps à longue portée, Ann. Inst. H. Poincaré, 59 (1993), 237-267.
D) J. Derezinski, Asymptotic completeness for N-particle long-range quantum systems, Ann, of Math., 138 (1993), 427-476.
FH) R. Froese and I. Herbst, Exponential bounds and absence of positive eigenvalues for N-body Schrödinger operators, Comm. Math. Phys., 87 (1982), 429-447.
Ge) C. Gérard, Sharp, Sharp propagation estimates for N-particle systems, Duke Math. J., 67 (1992), 483-515.
GIS) C. Gérard, H. Isozaki and E. Skibsted, Commutator algebra and resolvent estimates, Spectral and Scattering Theory and Applications, (ed. K. Yajima), Adv. Stud. Pure, Math., 23, 1994, pp. 69-82.
Gr) G. M. Graf, Asymptotic completeness for N-body short-range quantum systems: a new proof, Comm. Math. Phys., 132 (1990), 73-101.
HSj) B. Heiffer and J. Sjöstrand, Équation de Schrödinger avec champ magnétique et équation de Harper, Lecture Notes in Phys., 345, Springer, Berlin-Heidelberg-New York, 1989, pp. 118-197.
HSk) I. Herbst and E. Skibsted, Free channel Fourier transform in the long-range N-body problem, J. Analyse Math., 65 (1995), 297-332.
H1) L. Hörmander, The analysis of linear partial differential operators IV, Springer, Berlin-Heidelberg-New York, 1985.
H2) L. Hörmander, The analysis of linear partial differential operators I, Springer, Berlin-Heidelberg-New York, 1990.
I1) H. Isozaki, Differentiability of generalized Fourier transforms assciated with Schrödinger operators, J. Math. Kyoto Univ., 25-4 (1985), 789-806.
I2) H. Isozaki, Structures of S-matrices for three body Schrödinger operators, Comm. Math. Phys., 146 (1992), 241-258.
I3) H. Isozaki, Asymptotic properties of generalized eigenfunctions for three body Schrödinger operators, Comm. Math. Phys., 153 (1993), 1-21.
I4) H. Isozaki, A generalization of the radiation condition of Sommerfeld for N-body Schrödinger operators, Duke Math. J., 74 (1994), 557-584.
J) A. Jensen, Propagation estimates for Schrödinger-type operators, Trans. Amer. Math. Soc., 291-1 (1985), 129-144.
JMP) A. Jensen, E. Mourre and P. Perry, Multiple commutator estimates and resolvent smoothness in quantum scattering theory, Ann. Inst. H. Poincaré, 41-2 (1984), 207-225.
M1) E. Mourre, Absence of singular continuous spectrum for certain selfadjoint operators, Comm. Math. Phys., 78 (1981), 391-408.
M2) E. Mourre, Opérateurs conjugés et propriétés de propagations, Comm. Math. Phys., 91 (1983), 279-300.
P) P. Perry, Exponential bounds and semi-finiteness of point spectrum for N-body Schrödinger operators, Comm. Math. Phys., 92 (1984), 481-483.
PPS) P. Perry, I. M. Sigal and B. Simon, Spectral analysis of N-body Schrödinger operators, Ann. of Math., 114 (1981), 519-567.
SS) I. M. Sigal and A. Soffer, Local decay and propagation estimates for time dependent and time independent Hamiltonians, preprint, Princeton University, 1988.
S1) E. Skibsted, Smoothness of N-body scattering amplitudes, Reviews in Math. Phys., 4-4 (1992), 619-658.
S2) E. Skibsted, Propagation estimates for N-body Schrödinger operators, Comm. Math. Phys., 142 (1991), 67-98.
W1) X. P. Wang, On the three-body long-range scattering problems, Lett. Math. Phys., 25 (1992), 267-276.
W2) X. P. Wang, Microlocal resolvent estimates for N-body Schrödinger operators, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 40-2 (1993), 337-385.

Right : [B] A. Bommier, Propriétés de la matrice de diffusion, 2-auras-2-amas, pour les problèmes à N corps à longue portée, Ann. Inst. H. Poincaré, 59 (1993), 237-267.
[D] J. Derezinski, Asymptotic completeness for N-particle long-range quantum systems, Ann, of Math., 138 (1993), 427-476.
[FH] R. Froese and I. Herbst, Exponential bounds and absence of positive eigenvalues for N-body Schrödinger operators, Comm. Math. Phys., 87 (1982), 429-447.
[Ge] C. Gérard, Sharp propagation estimates for N-particle systems, Duke Math. J., 67 (1992), 483-515.
[GIS] C. Gérard, H. Isozaki and E. Skibsted, Commutator algebra and resolvent estimates, Spectral and Scattering Theory and Applications, (ed. K. Yajima), Adv. Stud. Pure, Math., 23, 1994, pp. 69-82.
[Gr] G. M. Graf, Asymptotic completeness for N-body short-range quantum systems: a new proof, Comm. Math. Phys., 132 (1990), 73-101.
[HSj] B. Heiffer and J. Sjöstrand, Équation de Schrödinger avec champ magnétique et équation de Harper, Lecture Notes in Phys., 345, Springer, Berlin-Heidelberg-New York, 1989, pp. 118-197.
[HSk] I. Herbst and E. Skibsted, Free channel Fourier transform in the long-range N-body problem, J. Analyse Math., 65 (1995), 297-332.
[H1] L. Hörmander, The analysis of linear partial differential operators IV, Springer, Berlin-Heidelberg-New York, 1985.
[H2] L. Hörmander, The analysis of linear partial differential operators I, Springer, Berlin-Heidelberg-New York, 1990.
[I1] H. Isozaki, Differentiability of generalized Fourier transforms assciated with Schrödinger operators, J. Math. Kyoto Univ., 25-4 (1985), 789-806.
[I2] H. Isozaki, Structures of S-matrices for three body Schrödinger operators, Comm. Math. Phys., 146 (1992), 241-258.
[I3] H. Isozaki, Asymptotic properties of generalized eigenfunctions for three body Schrödinger operators, Comm. Math. Phys., 153 (1993), 1-21.
[I4] H. Isozaki, A generalization of the radiation condition of Sommerfeld for N-body Schrödinger operators, Duke Math. J., 74 (1994), 557-584.
[J] A. Jensen, Propagation estimates for Schrödinger-type operators, Trans. Amer. Math. Soc., 291-1 (1985), 129-144.
[JMP] A. Jensen, E. Mourre and P. Perry, Multiple commutator estimates and resolvent smoothness in quantum scattering theory, Ann. Inst. H. Poincaré, 41-2 (1984), 207-225.
[M1] E. Mourre, Absence of singular continuous spectrum for certain selfadjoint operators, Comm. Math. Phys., 78 (1981), 391-408.
[M2] E. Mourre, Opérateurs conjugés et propriétés de propagations, Comm. Math. Phys., 91 (1983), 279-300.
[P] P. Perry, Exponential bounds and semi-finiteness of point spectrum for N-body Schrödinger operators, Comm. Math. Phys., 92 (1984), 481-483.
[PPS] P. Perry, I. M. Sigal and B. Simon, Spectral analysis of N-body Schrödinger operators, Ann. of Math., 114 (1981), 519-567.
[SS] I. M. Sigal and A. Soffer, Local decay and propagation estimates for time dependent and time independent Hamiltonians, preprint, Princeton University, 1988.
[S1] E. Skibsted, Smoothness of N-body scattering amplitudes, Reviews in Math. Phys., 4-4 (1992), 619-658.
[S2] E. Skibsted, Propagation estimates for N-body Schrödinger operators, Comm. Math. Phys., 142 (1991), 67-98.
[W1] X. P. Wang, On the three-body long-range scattering problems, Lett. Math. Phys., 25 (1992), 267-276.
[W2] X. P. Wang, Microlocal resolvent estimates for N-body Schrödinger operators, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 40-2 (1993), 337-385.

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