Inverse scattering for singular potentials in two dimensions

Sun, Zi Qi; Uhlmann, Gunther

doi:10.1090/S0002-9947-1993-1126214-4

Inverse scattering for singular potentials in two dimensions
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by Zi Qi Sun and Gunther Uhlmann PDF

Trans. Amer. Math. Soc. 338 (1993), 363-374 Request permission

Abstract:

We consider the Schrödinger equation for a compactly supported potential having jump type singularities at a subdomain of ${\mathbb {R}^2}$. We prove that knowledge of the scattering amplitude at a fixed energy, determines the location of the singularity as well as the jump across the curve of discontinuity. This result follows from a similar result for the Dirichlet to Neumann map associated to the Schrödinger equation for a compactly supported potential with the same type of singularities.

References

Yu. M. Berezanski, The uniqueness theorem in the inverse problem of spectral analysis for the Schrödinger equation, Trudy Moskov. Mat. Obšč. 7 (1958), 1–62 (Russian). MR 0101377
R. Beals and R. R. Coifman, Multidimensional inverse scatterings and nonlinear partial differential equations, Pseudodifferential operators and applications (Notre Dame, Ind., 1984) Proc. Sympos. Pure Math., vol. 43, Amer. Math. Soc., Providence, RI, 1985, pp. 45–70. MR 812283, DOI 10.1090/pspum/043/812283
Victor Isakov, On uniqueness in the inverse transmission scattering problem, Comm. Partial Differential Equations 15 (1990), no. 11, 1565–1587. MR 1079603, DOI 10.1080/03605309908820737
Adrian I. Nachman, Reconstructions from boundary measurements, Ann. of Math. (2) 128 (1988), no. 3, 531–576. MR 970610, DOI 10.2307/1971435
Adrian I. Nachman and Mark J. Ablowitz, A multidimensional inverse-scattering method, Stud. Appl. Math. 71 (1984), no. 3, 243–250. MR 769078, DOI 10.1002/sapm1984713243
Adrian Nachman, John Sylvester, and Gunther Uhlmann, An $n$-dimensional Borg-Levinson theorem, Comm. Math. Phys. 115 (1988), no. 4, 595–605. MR 933457
R. G. Novikov, A multidimensional inverse spectral problem for the equation $-\Delta \psi +(v(x)-Eu(x))\psi =0$, Funktsional. Anal. i Prilozhen. 22 (1988), no. 4, 11–22, 96 (Russian); English transl., Funct. Anal. Appl. 22 (1988), no. 4, 263–272 (1989). MR 976992, DOI 10.1007/BF01077418
Zi Qi Sun, On an inverse boundary value problem in two dimensions, Comm. Partial Differential Equations 14 (1989), no. 8-9, 1101–1113. MR 1017066, DOI 10.1080/03605308908820646
Zi Qi Sun, The inverse conductivity problem in two dimensions, J. Differential Equations 87 (1990), no. 2, 227–255. MR 1072901, DOI 10.1016/0022-0396(90)90002-7

Generic uniqueness for determined inverse problems in

dimensions

Zi Qi Sun and Gunther Uhlmann, Generic uniqueness for an inverse boundary value problem, Duke Math. J. 62 (1991), no. 1, 131–155. MR 1104326, DOI 10.1215/S0012-7094-91-06206-X
Zi Qi Sun and Gunther Uhlmann, Recovery of singularities for formally determined inverse problems, Comm. Math. Phys. 153 (1993), no. 3, 431–445. MR 1218927
John Sylvester and Gunther Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2) 125 (1987), no. 1, 153–169. MR 873380, DOI 10.2307/1971291
John Sylvester and Gunther Uhlmann, A uniqueness theorem for an inverse boundary value problem in electrical prospection, Comm. Pure Appl. Math. 39 (1986), no. 1, 91–112. MR 820341, DOI 10.1002/cpa.3160390106