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Recovery of singularities for formally determined inverse problems. (English) Zbl 0795.35142

This paper deals with the Schrödinger equation in \(\mathbb{R}^ 2\) for potentials of compact support in \(L^ \infty\), and with the acoustic equation in \(\mathbb{R}^ 2\) with a bounded obstacle in which the sound speed is in \(L^ \infty\). The authors prove that if the scattering amplitude for the Schrödinger equation is given for all angles and one energy, the potential is determined to within a function in \(C^ \alpha(\mathbb{R}^ 2)\), \(\forall\alpha\), \(0\leq \alpha<1\), where \(C^ \alpha (\mathbb{R}^ 2)\) is the Hölder space of order \(\alpha\). Furthermore, the potential (again to within the same possible ambiguity) is equal to the inverse Fourier transform of the function \(T_ q\) used by Beals and Coifman and by Ablowitz and Nachman in the \(\overline{\partial}\) approach. They then rely on the fact that the Dirichlet-to-Neumann map determines \(T_ q\) to suggest that the location and strengths of the singularities of \(q\) can be obtained from that map by the method of Nachman.
Similarly, the authors prove that if the values of the fundamental solutions of the acoustic equation on the surface of the obstacle are given, the sound speed is determined to within a function in \(C^ \alpha(\mathbb{R}^ 2)\), \(\forall\alpha\), \(0\leq\alpha<1\).

MSC:

35R30 Inverse problems for PDEs
35P25 Scattering theory for PDEs
35Q40 PDEs in connection with quantum mechanics
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[1] [A-S] Alsholm, P., Schmidt, G.: Spectral and scattering theory for Schrödinger operators. Archive Rat. Mech.40, 281–311 (1971) · Zbl 0226.35076
[2] [B-C] Beals, R., Coifman, R.: Multidimensional inverse scattering and nonlinear PDE. Proc. Symp. Pure Math.43, Providence: Am. Math. Soc. 45–70 (1985)
[3] [I] Ikehata, M.: A special Green’s function for the biharmonic operator and its application to an inverse boundary value problem. Comp. Math. Appl.22, 53–66 (1991) · Zbl 0770.35078 · doi:10.1016/0898-1221(91)90131-M
[4] [K-V] Kohn, R., Vogelius, M.: Determining conductivity by boundary measurements II, Interior results. Comm. Pure Appl. Math.XXXVIII, 643–667 (1985) · Zbl 0595.35092 · doi:10.1002/cpa.3160380513
[5] [N] Nachman, A.: Reconstructions from boundary measurements. Ann. Math.128, 531–587 (1988) · Zbl 0675.35084 · doi:10.2307/1971435
[6] [N-A] Nachman, A., Ablowitz, M.: A multidimensional inverse scattering method. Studies in Appl. Math.71, 243–250 (1984) · Zbl 0557.35032
[7] [N-W] Nirenberg, L., Walker, H.: Null spaces of elliptic partial differential operators in \(\mathbb{R}\) n . J. Math. Anal. Appl.42, 271–301 (1973) · Zbl 0272.35029 · doi:10.1016/0022-247X(73)90138-8
[8] [St] Stefanov, P.: Stability of the inverse problem in potential scattering at fixed energy. Ann. Inst. F.40, 867–888 (1990) · Zbl 0715.35082
[9] [Su I] Sun, Z.: On an inverse boundary value problem in two dimensions. Comm. P.D.E.14, 1101–1113 (1989) · Zbl 0704.35132 · doi:10.1080/03605308908820646
[10] [Su II] Sun, Z.: The inverse conductivity problem in two dimensions. J. Diff. Eq.87, 227–255 (1990) · Zbl 0716.35080 · doi:10.1016/0022-0396(90)90002-7
[11] [Su-U I] Sun, Z., Uhlmann, G.: Generic uniqueness for an inverse boundary value problem. Duke Math. J.62, 131–155 (1991) · Zbl 0728.35132 · doi:10.1215/S0012-7094-91-06206-X
[12] [Su-U II] Sun, Z., Uhlmann, G.: Inverse scattering for singular potentials in two dimensions. to appear Trans. AMS.
[13] [S-U] Sylvester, J., Uhlmann, G.: A uniqueness theorem for an inverse boundary value problem in electrical prospection. Comm. Pure Appl. Math.XXXIX, 91–112 (1986) · Zbl 0611.35088 · doi:10.1002/cpa.3160390106
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