On solutions of the Schrödinger equation with radiation conditions at infinity : the long-range case

Yannick Gâtel; Dimitri Yafaev

Annales de l'institut Fourier (1999)

  • Volume: 49, Issue: 5, page 1581-1602
  • ISSN: 0373-0956

Abstract

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We consider the homogeneous Schrödinger equation with a long-range potential and show that its solutions satisfying some a priori bound at infinity can asymptotically be expressed as a sum of incoming and outgoing distorted spherical waves. Coefficients of these waves are related by the scattering matrix. This generalizes a similar result obtained earlier in the short-range case.

How to cite

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Gâtel, Yannick, and Yafaev, Dimitri. "On solutions of the Schrödinger equation with radiation conditions at infinity : the long-range case." Annales de l'institut Fourier 49.5 (1999): 1581-1602. <http://eudml.org/doc/75394>.

@article{Gâtel1999,
abstract = {We consider the homogeneous Schrödinger equation with a long-range potential and show that its solutions satisfying some a priori bound at infinity can asymptotically be expressed as a sum of incoming and outgoing distorted spherical waves. Coefficients of these waves are related by the scattering matrix. This generalizes a similar result obtained earlier in the short-range case.},
author = {Gâtel, Yannick, Yafaev, Dimitri},
journal = {Annales de l'institut Fourier},
keywords = {scattering theory; long-range potentials; homogeneous Schrödinger equation; class of solutions; scattering matrix},
language = {eng},
number = {5},
pages = {1581-1602},
publisher = {Association des Annales de l'Institut Fourier},
title = {On solutions of the Schrödinger equation with radiation conditions at infinity : the long-range case},
url = {http://eudml.org/doc/75394},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Gâtel, Yannick
AU - Yafaev, Dimitri
TI - On solutions of the Schrödinger equation with radiation conditions at infinity : the long-range case
JO - Annales de l'institut Fourier
PY - 1999
PB - Association des Annales de l'Institut Fourier
VL - 49
IS - 5
SP - 1581
EP - 1602
AB - We consider the homogeneous Schrödinger equation with a long-range potential and show that its solutions satisfying some a priori bound at infinity can asymptotically be expressed as a sum of incoming and outgoing distorted spherical waves. Coefficients of these waves are related by the scattering matrix. This generalizes a similar result obtained earlier in the short-range case.
LA - eng
KW - scattering theory; long-range potentials; homogeneous Schrödinger equation; class of solutions; scattering matrix
UR - http://eudml.org/doc/75394
ER -

References

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  11. [11] R. MELROSE, M. ZWORSKI, Scattering metrics and geodesic flow at infinity, Invent. Math., 124 (1996), 389-436. Zbl0855.58058MR96k:58230
  12. [12] Y. SAITO, Spectral representations for Schrödinger operators with long-range potentials, Lecture Notes in Math., 727, Springer, Berlin, 1979. Zbl0414.47012MR81a:35083
  13. [13] Y. SAITO, On the S-matrix for Schrödinger operators with long-range potentials, J. reine Angew. Math., 314 (1980), 99-116. Zbl0422.35024MR82h:35083
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  16. [16] K. YOSIDA, Functional analysis, Springer-Verlag, 1966. Zbl0126.11504

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