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Scattering problems for perturbations of the multidimensional biharmonic operator
Parametrices for the light ray transform on Minkowski spacetime
Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195-4350, USA |
We consider restricted light ray transforms arising from an inverse problem of finding cosmic strings. We construct a relative left parametrix for the transform on two tensors, which recovers the space-like and some light-like singularities of the two tensor.
References:
[1] |
J. Antoniano and G. Uhlmann,
A functional calculus for a class of pseudodifferential operators with singular symbols, Proc. Symp. Pure Math., 43 (1985), 5-16.
|
[2] |
M. de Hoop, G. Uhlmann and A. Vasy,
Diffraction from conormal singularities, Annales Scientifiques de l'École Normale Supérieure, 4e serie, 48 (2015), 351-408.
doi: 10.24033/asens.2247. |
[3] |
A. Greenleaf and A. Seeger,
Fourier integral operators with fold singularities, J. reine angew. Math., 455 (1994), 35-56.
|
[4] |
A. Greenleaf and G. Uhlmann,
Nonlocal inversion formulas for the X-ray transform, Duke Math. J., 58 (1989), 205-240.
doi: 10.1215/S0012-7094-89-05811-0. |
[5] |
A. Greenleaf and G. Uhlmann,
Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms, Annales de l'institut Fourier, 40 (1990), 443-466.
doi: 10.5802/aif.1220. |
[6] |
A. Greenleaf and G. Uhlmann,
Estimates for singular Radon transforms and pseudodifferential operators with singular symbols, Journal of Functional Analysis, 89 (1990), 202-232.
doi: 10.1016/0022-1236(90)90011-9. |
[7] |
A. Greenleaf and G. Uhlmann,
Microlocal techniques in integral geometry, Contemporary Mathematics, 113 (1990), 121-135.
|
[8] |
A. Greenleaf and G. Uhlmann,
Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms. Ⅱ, Duke Math. J., 64 (1991), 415-444.
doi: 10.1215/S0012-7094-91-06422-7. |
[9] |
A. Greenleaf and G. Uhlmann,
Recovering singularities of a potential from singularities of scattering data, Communications in Mathematical Physics, 157 (1993), 549-572.
doi: 10.1007/BF02096882. |
[10] |
V. Guillemin,
Cosmology in $(2+1) $-Dimensions, Cyclic Models, and Deformations of $M_{2, 1} $ Annals of Mathematics Studies, No. 121, Princeton University Press, 1989. |
[11] |
V. Guillemin and G. Uhlmann,
Oscillatory integrals with singular symbols, Duke Math. J., 48 (1981), 251-267.
doi: 10.1215/S0012-7094-81-04814-6. |
[12] |
L. Hörmander,
Fourier integral operators. Ⅰ, Acta Mathematica, 127 (1971), 79-183.
doi: 10.1007/BF02392052. |
[13] |
L. Hörmander,
The Analysis of Linear Partial Differential Operators Ⅳ: Fourier Integral Operators Springer-Verlag, Berlin, Heidelberg, 2009. |
[14] |
M. Lassas, L. Oksanen, P. Stefanov and G. Uhlmann, On the inverse problem of finding cosmic strings and other topological defects, preprint, arXiv: 1505.03123. |
[15] |
R. Melrose and G. Uhlmann,
Lagrangian intersection and the Cauchy problem, Communications on Pure and Applied Mathematics, 32 (1979), 483-519.
doi: 10.1002/cpa.3160320403. |
[16] |
B. Palacios, G. Uhlmann and Y. Wang,
Reducing streaking artifacts in quantitative susceptibility mapping, SIAM Journal of Imaging Sciences, 10 (2017), 1921-1934.
|
[17] |
P. Stefanov, Support theorems for the light ray transform on analytic Lorentzian manifolds, Proc. Amer. Math. Soc., 145 (2017), 1259–1274. arXiv: 1504.01184. |
show all references
References:
[1] |
J. Antoniano and G. Uhlmann,
A functional calculus for a class of pseudodifferential operators with singular symbols, Proc. Symp. Pure Math., 43 (1985), 5-16.
|
[2] |
M. de Hoop, G. Uhlmann and A. Vasy,
Diffraction from conormal singularities, Annales Scientifiques de l'École Normale Supérieure, 4e serie, 48 (2015), 351-408.
doi: 10.24033/asens.2247. |
[3] |
A. Greenleaf and A. Seeger,
Fourier integral operators with fold singularities, J. reine angew. Math., 455 (1994), 35-56.
|
[4] |
A. Greenleaf and G. Uhlmann,
Nonlocal inversion formulas for the X-ray transform, Duke Math. J., 58 (1989), 205-240.
doi: 10.1215/S0012-7094-89-05811-0. |
[5] |
A. Greenleaf and G. Uhlmann,
Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms, Annales de l'institut Fourier, 40 (1990), 443-466.
doi: 10.5802/aif.1220. |
[6] |
A. Greenleaf and G. Uhlmann,
Estimates for singular Radon transforms and pseudodifferential operators with singular symbols, Journal of Functional Analysis, 89 (1990), 202-232.
doi: 10.1016/0022-1236(90)90011-9. |
[7] |
A. Greenleaf and G. Uhlmann,
Microlocal techniques in integral geometry, Contemporary Mathematics, 113 (1990), 121-135.
|
[8] |
A. Greenleaf and G. Uhlmann,
Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms. Ⅱ, Duke Math. J., 64 (1991), 415-444.
doi: 10.1215/S0012-7094-91-06422-7. |
[9] |
A. Greenleaf and G. Uhlmann,
Recovering singularities of a potential from singularities of scattering data, Communications in Mathematical Physics, 157 (1993), 549-572.
doi: 10.1007/BF02096882. |
[10] |
V. Guillemin,
Cosmology in $(2+1) $-Dimensions, Cyclic Models, and Deformations of $M_{2, 1} $ Annals of Mathematics Studies, No. 121, Princeton University Press, 1989. |
[11] |
V. Guillemin and G. Uhlmann,
Oscillatory integrals with singular symbols, Duke Math. J., 48 (1981), 251-267.
doi: 10.1215/S0012-7094-81-04814-6. |
[12] |
L. Hörmander,
Fourier integral operators. Ⅰ, Acta Mathematica, 127 (1971), 79-183.
doi: 10.1007/BF02392052. |
[13] |
L. Hörmander,
The Analysis of Linear Partial Differential Operators Ⅳ: Fourier Integral Operators Springer-Verlag, Berlin, Heidelberg, 2009. |
[14] |
M. Lassas, L. Oksanen, P. Stefanov and G. Uhlmann, On the inverse problem of finding cosmic strings and other topological defects, preprint, arXiv: 1505.03123. |
[15] |
R. Melrose and G. Uhlmann,
Lagrangian intersection and the Cauchy problem, Communications on Pure and Applied Mathematics, 32 (1979), 483-519.
doi: 10.1002/cpa.3160320403. |
[16] |
B. Palacios, G. Uhlmann and Y. Wang,
Reducing streaking artifacts in quantitative susceptibility mapping, SIAM Journal of Imaging Sciences, 10 (2017), 1921-1934.
|
[17] |
P. Stefanov, Support theorems for the light ray transform on analytic Lorentzian manifolds, Proc. Amer. Math. Soc., 145 (2017), 1259–1274. arXiv: 1504.01184. |

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