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On uniqueness in the inverse transmission scattering problem. (English) Zbl 0728.35148
In this paper uniqueness results are proved for the inverse scattering problems where the unknown scatterer D is a bounded open set and some coefficients of an elliptic equation are unknown as well. Let \(D^ i\) be the bounded open set in \(R^ n\), \(D^ e=R^ n\setminus D^ i\), \(u=u^ e\) in \(D^ e\), \(u=u^ i\) in \(D^ i\), \(\chi\) (D) the characteristic function of D, \(a=1+(\mu -1)\chi (D)\), \(c=1+(\rho -1)\chi (D)\). Further, let u be a solution of \(div(a\nabla u)+k^ 2cu=0\), satisfying \(u^ i=u^ e\), \(\partial u^ e/\partial N=\mu \partial u^ i/\partial N\) on \(\partial D\), \(u^ e(x)=\exp (ix\cdot \xi +u^{e_ 0}(x)\), \(| \xi | =k\), \(r^{(n-1)/2}(\partial u^{e_ 0}/\partial r-iku^{e_ 0})\to 0\) for \(r\to \infty\). The author applies ideas of Nachman, Sylvester, Uhlmann and own results for this special problem under consideration.
Reviewer: G.Anger (Berlin)

MSC:
35R30 Inverse problems for PDEs
35P25 Scattering theory for PDEs
35J10 Schrödinger operator, Schrödinger equation
78A40 Waves and radiation in optics and electromagnetic theory
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References:
[1] Angell T. S., Inverse Problems 3 pp 149– (1987) · Zbl 0658.35093
[2] Colton D., Integral Equations Methods is Scattering Theory (1983)
[3] Isakov V., Soviet Math. Dokl 17 pp 1338– (1976)
[4] Isakov V., Comm. Pure Appl. Math. 41 pp 865– (1988) · Zbl 0676.35082
[5] V. Isakov Completeness of products and inverse problems for PDE] J. of Diff. Equations (in Print).
[6] Isakov V., AMS math.
[7] Lax P., Scattering Theory (1967)
[8] Lax P., Comm. Pure Appl. Math. 30 pp 195– (1977) · Zbl 0335.35075
[9] Majda A., Comm. Pure Appl. Math. 30 pp 169– (1977)
[10] Taylor M., Comm. in part. Diff. Equat. 2 (4) pp 395– (1977) · Zbl 0357.35010
[11] Melrose R., Comm. Pure Appl. Math. 33 (4) pp 461– (1980) · Zbl 0435.35066
[12] Miranda C., Ergebn. der Math. (1970)
[13] Morrey C. B., Die Grundlehren der Math. 130 (1966)
[14] Morrey C. B., Ann. Math. 128 pp 531– (1988) · Zbl 0675.35084
[15] Nachman A., Comm. Math. Physics 115 pp 595– (1973) · Zbl 0644.35095
[16] Phillips R., Proc. Symp. in Pure Math. 130 pp 153– (1966)
[17] Stampacchia G., :Les Presses de l’Univ. de Montreal 23 (1966)
[18] Sylvester J., Ann. Math. 125 pp 153– (1987) · Zbl 0625.35078
[19] Sylvester J., SIAM Proc. Series 125 pp 99– (1990)
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