Recovering the total singularity of a conormal potential from backscattering data

Mark S. Joshi

Annales de l'institut Fourier (1998)

  • Volume: 48, Issue: 5, page 1513-1532
  • ISSN: 0373-0956

Abstract

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The problem of recovering the singularities of a potential from backscattering data is studied. Let Ω be a smooth precompact domain in ℝ n which is convex (or normally accessible). Suppose V i = v + w i with v ∈ C c ∞ ( ℝ n ) and w i conormal to the boundary of Ω and supported inside Ω ‾ then if the backscattering data of V 1 and V 2 are equal up to smoothing, we show that w 1 - w 2 is smooth.

How to cite

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Joshi, Mark S.. "Recovering the total singularity of a conormal potential from backscattering data." Annales de l'institut Fourier 48.5 (1998): 1513-1532. <http://eudml.org/doc/75328>.

@article{Joshi1998,
abstract = {The problem of recovering the singularities of a potential from backscattering data is studied. Let $\Omega $ be a smooth precompact domain in $\{\Bbb R\}^n$ which is convex (or normally accessible). Suppose $V_i = v + w_i$ with $v \in C^\{\infty \}_\{c\}(\{\Bbb R\}^n)$ and $w_i$ conormal to the boundary of $\Omega $ and supported inside $\bar\{\Omega \}$ then if the backscattering data of $V_1$ and $V_2$ are equal up to smoothing, we show that $w_1 - w_2$ is smooth.},
author = {Joshi, Mark S.},
journal = {Annales de l'institut Fourier},
keywords = {inverse scattering problem; backscattering; conormal potential; Lagrangian distributions; Lax-Phillips scattering theory},
language = {eng},
number = {5},
pages = {1513-1532},
publisher = {Association des Annales de l'Institut Fourier},
title = {Recovering the total singularity of a conormal potential from backscattering data},
url = {http://eudml.org/doc/75328},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Joshi, Mark S.
TI - Recovering the total singularity of a conormal potential from backscattering data
JO - Annales de l'institut Fourier
PY - 1998
PB - Association des Annales de l'Institut Fourier
VL - 48
IS - 5
SP - 1513
EP - 1532
AB - The problem of recovering the singularities of a potential from backscattering data is studied. Let $\Omega $ be a smooth precompact domain in ${\Bbb R}^n$ which is convex (or normally accessible). Suppose $V_i = v + w_i$ with $v \in C^{\infty }_{c}({\Bbb R}^n)$ and $w_i$ conormal to the boundary of $\Omega $ and supported inside $\bar{\Omega }$ then if the backscattering data of $V_1$ and $V_2$ are equal up to smoothing, we show that $w_1 - w_2$ is smooth.
LA - eng
KW - inverse scattering problem; backscattering; conormal potential; Lagrangian distributions; Lax-Phillips scattering theory
UR - http://eudml.org/doc/75328
ER -

References

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  8. [8] M.S. JOSHI, A Precise Calculus of Paired Lagrangian Distributions, M.I.T. thesis, 1994. 
  9. [9] M.S. JOSHI, A Symbolic Contruction of the Forward Fundamental Solution of the Wave Operator, preprint. Zbl0917.35171
  10. [10] M.S. JOSHI and A. SA BARRETO, Recovering Asymptotics of Short Range Potentials, to appear in Commun. in Math. Phys. Zbl0920.58052
  11. [11] P. LAX, R. PHILLIPS, Scattering Theory, Revised Edition. New York, London: Academic Press, 1989. Zbl0697.35004MR90k:35005
  12. [12] R.B. MELROSE, Differential Analysis on Manifolds with Corners, forthcoming. 
  13. [13] R.B. MELROSE, Marked Lagrangian Distributions, manuscript. 
  14. [14] R.B. MELROSE and G. UHLMANN, Lagrangian Intersection and the Cauchy Problem, Comm. on Pure and Applied Math., 32 (1979), 482-512. Zbl0396.58006MR81d:58052
  15. [15] R. PHILIPS, Scattering Theory for the Wave Equation with Short Range Potential, Indiana Univ. Math. Journal, 31 (1982), 609-639. Zbl0465.35073

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