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Recovering singularities of a potential from singularities of scattering data. (English) Zbl 0790.35112
The authors consider recovery of a compactly supported potential $$q$$ on $$\mathbb{R}^ n$$, $$n\geq 3$$, of the Schrödinger equation $$-\Delta+q$$ from the backscattering data $$A(\omega, -\omega,\lambda)$$, $$\omega\in S^{n- 1}$$, $$\lambda\in\mathbb{R}$$, where $$A$$ is the scattering amplitude (far field pattern) of this Schrödinger operator.
Recently G. Eskin and J. Ralston [Commun. Math. Phys. 124, No. 2, 169-215 (1989; Zbl 0706.35136)] proved that the map from $$q$$ to the backscattering data is a locally invertible for generic (“almost all”) $$q$$, however still there is no global uniqueness result for $$q$$. By using the associated wave equation and microlocal analysis the authors prove uniqueness of recovery of a smooth surface $$S$$ and normal jumps of $$q$$ and its derivatives across $$S$$.
Reviewer: V.Isakov (Wichita)

MSC:
 35R30 Inverse problems for PDEs 35P25 Scattering theory for PDEs 81U40 Inverse scattering problems in quantum theory
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References:
 [1] [BLM] Bayliss, A., Lin, Y., Morawetz, C.: Scattering by a potential using hyperbolic methods. Math. Comp.52, 321–338 (1989) · Zbl 0692.65068 · doi:10.1090/S0025-5718-1989-0958869-1 [2] [BC] Beals, R., Coifman, R.: Multidimensional inverse scattering and nonlinear PDE. Proc. Symp. Pure Math.43, 45–70 (1985) [3] [DG] Duistermaat, J.J., Guillemin, V.: The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math.29, 39–79 (1975) · Zbl 0307.35071 · doi:10.1007/BF01405172 [4] [DH] Duistermaat, J.J., Hörmander, L.: Fourier integral operators, II. Acta. Math.128, 183–269 (1972) · Zbl 0232.47055 · doi:10.1007/BF02392165 [5] [ER] Eskin, G., Ralston, J.: The inverse backscattering problem in three dimensions. Commun. Math. Phys.124, 169–215 (1989) · Zbl 0706.35136 · doi:10.1007/BF01219194 [6] [GU] Greenleaf, A., Uhlmann, G.: Estimates for singular Radon transforms and pseudodifferential operators with singular symbols. J. Funct. Anal.89, 202–232 (1990) · Zbl 0717.44001 · doi:10.1016/0022-1236(90)90011-9 [7] [GuU] Guillemin, V., Uhlmann, G.: Oscillatory integrals with singular symbols. Duke Math. J.48, 251–267 (1981) · Zbl 0462.58030 · doi:10.1215/S0012-7094-81-04814-6 [8] [HN] Henkin, G., Novikov, R.: A multidimensional inverse problem in quantum and acoustic scattering. Inverse Prob.4, 103–121 (1988) · Zbl 0697.35108 · doi:10.1088/0266-5611/4/1/011 [9] [Hö] Hörmander, L.: Fourier integral operators, I. Acta Math.127, 79–183 (1971) · Zbl 0212.46601 · doi:10.1007/BF02392052 [10] [LP] Lax, P., Phillips, R.: Scattering Theory. Revised Edition. New York, London: Academic Press 1989 [11] [MU1] Melrose, R., Uhlmann, G.: Lagrangian intersection and the Cauchy problem. Comm. Pure Appl. Math.32, 482–512 [12] [MU2] Melrose, R., Uhlmann, G.: Introduction to Microlocal Analysis with Applications to Scattering Theory. Book in preparation [13] [N] Nachman, A.: Inverse Scattering at fixed energy. Proc. Int. Cong. Math. Phys. Leipzig, 1991 (to appear) · Zbl 0947.81562 [14] [No] Novikov, R.: Multidimensional inverse spectral problem for the equation -$$\Delta$$$$\psi$$+(V(x)-Eu(x))$$\psi$$=0. Funct. Anal. Appl.22, 263–272 (1988) · Zbl 0689.35098 · doi:10.1007/BF01077418 [15] [Pe] Petkov, V.: Scattering theory for hyperbolic operators. Amsterdam: North-Holland 1989 · Zbl 0687.35067 [16] [P] Phillips, R.: Scattering theory for the wave equation with a short range potential. Indiana Univ. Math. J.31, 609–639 (1982) · Zbl 0489.35066 · doi:10.1512/iumj.1982.31.31045 [17] [PäS] Päivärinta, L., Somersalo, E.: Inversion of discontinuities for the Schrödinger operator in three dimensions. SIAM J. Math. Anal.22, 480–499 (1991) · Zbl 0731.35095 · doi:10.1137/0522031 [18] [Pi] Piriou, A.: Calcul symbolique non lineare pour une onde conormale simple. Ann. Inst. Four.38, 173–188 (1988) · Zbl 0646.35012 [19] [Pr] Prosser, R.J.: Formal solutions of inverse scattering problems, III and IV. J. Math. Phys.21, 2648–2653 (1980) and23, 2127–2130 (1982) · Zbl 0446.35077 · doi:10.1063/1.524379
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