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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Propagation estimates for Schrödinger-type operators

Author: Arne Jensen
Journal: Trans. Amer. Math. Soc. 291 (1985), 129-144
MSC: Primary 35P99; Secondary 81E13, 81F05
MathSciNet review: 797050
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Abstract: Propagation estimates for a Schrödinger-type operator are obtained using multiple commutator techniques. A new method is given for obtaining estimates for powers of the resolvent. As an application, micro-local propagation estimates are obtained for two-body Schrödinger operators with smooth long-range potentials.

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  • [1] S. Agmon, Some new results on spectral and scattering theory of differential operators on $ {{\mathbf{R}}^n}$, Séminaire Goulaouic-Schwartz (1978/79), Exposé no. 2, École Polytech., Palaiseau, 1979. MR 557513 (81j:35091)
  • [2] M. Combescure, Propagation and local-decay properties for long-range scattering of three-body systems, preprint, LPTHE, Orsay, 1984.
  • [3] H. Cycon and P. Perry, Local time-decay of high energy scattering states for the Schrödinger equation, preprint, 1983. MR 767370 (86b:35164)
  • [4] V. Enss, Propagation properties of quantum scattering states, J. Funct. Anal. 52 (1983), 219-251. MR 707205 (85b:81187)
  • [5] -, Completeness of three-body quantum scattering, Dynamics and Processes (P. Blanchard, L. Streit, eds.), Lecture Notes in Math., vol. 1031, Springer-Verlag, Berlin and New York, 1984.
  • [6] G. Hardy, J. Littlewood and G. Polya, Inequalities, 2nd ed., Cambridge Univ. Press, Cambridge, 1952. MR 0046395 (13:727e)
  • [7] L. Hörmander, The Weyl calculus of pseudo-differential operators, Comm. Pure Appl. Math. 32 (1979), 359-443.
  • [8] H. Isozaki, Differentiability of generalized Fourier transforms associated with Schrödinger equations, preprint, 1984.
  • [9] H. Isozaki and H. Kitada, Micro-local resolvent estimates for two-body Schrödinger operators, preprint, 1983.
  • [10] -, A remark on the micro-local resolvent estimates for two-body Schrödinger operators, preprint, 1983.
  • [11] A. Jensen, E. Mourre and P. Perry, Multiple commutator estimates and resolvent smoothness in quantum scattering theory, Ann. Inst. H. Poincaré, Sect. A (N.S) 41 (1984), 207-225. MR 769156 (86j:35126)
  • [12] E. Mourre, Link between the geometrical and spectral transformation approaches in scattering theory, Comm. Math. Phys. 68 (1979), 91-94. MR 539739 (82d:81127)
  • [13] -, Absence of singular continuous spectrum for certain self-adjoint operators, Comm. Math. Phys. 78 (1981), 391-408.
  • [14] -, Algebraic approach to some propagation properties of the Schrödinger equation, Mathematical Problems of Theoretical Physics (R. Schrader, R. Seiler, P. A. Uhlenbrock, eds.), Lecture Notes in Physics, vol. 153, Springer-Verlag, Berlin, 1981.
  • [15] -, Opérateurs conjugués et propriétés de propagation. I, II, Preprints, CPT-CNRS, Marseille, 1981, p. 1333; 1982, p. 1379.
  • [16] -, Opérateurs conjugués et propriétés de propagation, Comm. Math. Phys. 91 (1983), 279-300. MR 723552 (86h:47031)
  • [17] -, Private communication.
  • [18] P. Perry, Mellin transforms and scattering theory. I, Short range potentials, Duke Math. J. 47 (1980), 187-193. MR 563375 (81c:35101)
  • [19] -, Propagation of states in dilation analytic potentials and asymptotic completeness, Comm. Math. Phys. 81 (1981), 243-259. MR 632760 (84f:81097)
  • [20] P. Perry, I. Sigal and B. Simon, Spectral analysis of $ N$-body Schrödinger operators, Ann. of Math. (2) 114 (1981), 519-567. MR 634428 (83b:81129)
  • [21] M. Reed and B. Simon, Methods of modern mathematical physics. IV, Analysis of Operators, Academic Press, New York, 1978. MR 0493421 (58:12429c)
  • [22] M. Taylor, Pseudo-differential operators, Princeton Univ. Press, Princeton, N. J., 1981.

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