Inverse backscattering in two dimensions. (English) Zbl 0728.35146

Eskin, G.; Ralston, J.

Commun. Math. Phys. 138, No. 3, 451-486 (1991).

The authors extend their previous work [ibid. 124, No.2, 169-215 (1989; Zbl 0706.35136)] on inverse backscattering in three dimensions to two dimensional case. The new difficulty arises from the zero energy behaviour of backscattering amplitude, which, generically, has the form: \[ a(\xi /| \xi |,-\xi /| \xi |,| \xi |)=2\pi (2\pi \beta +\ln | \xi |)^{-1}+b(\xi), \] where \(b(0)=0\) and b(\(\xi\)) is Hölder \(\alpha\)-continuous, \(0<\alpha <1\). Let \(H_{\alpha,N}\) denote the weighted Hölder space. The authors prove that the mapping to the backscattering data is a local analytical diffeomorphism of \(H_{\alpha,N}\) on a dense open set of \(H_{\alpha,N}\), for \(0<\alpha <1/2\), and \(N>0\).

Reviewer: X.-P.Wang (Nantes)

MSC:

35R30	Inverse problems for PDEs
35P25	Scattering theory for PDEs
35Q60	PDEs in connection with optics and electromagnetic theory

Keywords:

inverse backscattering; backscattering amplitude

Citations:

Zbl 0706.35136

References:

[1]	[BC] Beals, R., Coifman, R.: Multidimensional inverse scattering and nonlinear partial differential equations. Proc. Symp. Pure Math.43, 45–70. Providence, RI: AMS, 1985
[2]	[BML] Bayliss, A., Li, Y., Morawetz, C.: Scattering by potentials using hyperbolic methods. Math. Comp.52, 321–338 (1989) · Zbl 0692.65068 · doi:10.1090/S0025-5718-1989-0958869-1
[3]	[C] Cheney, M.: Inverse scattering in dimension 2. J. Math. Phys.25, 94–107 (1984) · Zbl 0554.35126 · doi:10.1063/1.526003
[4]	[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
[5]	[F] Faddeev, L. D.: Inverse problem of quantum scattering theory, II. J. Sov. Math.5, 334–396 (1976) · Zbl 0373.35014 · doi:10.1007/BF01083780
[6]	[M] Melin, A.: The Lippman-Schwinger equation treated as a characteristic Cauchy problem. Seminaire sur EDP 1988–1989. Exposé IV. Ecole Polytech. Palaiseau · Zbl 0702.35065
[7]	[MU] Melrose, R., Uhlmann, G.: Introduction to microlocal analysis with applications to scattering theory, preprint 1989
[8]	[NA] Nachman, A., Ablowitz, M.: A multidimensional inverse scattering method. Stud. Appl. Math.71, 243–250 (1984) · Zbl 0557.35032
[9]	[N] Newton, R.: Inverse scattering II, III, IV. J. Math. Phys.21, 1968–1715 (1980); J. Math. Phys.22, 2191–2200 1981 (Correction, J. Math. Phys.23, 693 (1982)) J. Math. Phys.23, 594–604 (1982)
[10]	[NK] Novikov, R. G., Khenkin, G. M.: The \(\bar \partial \) -equation in multidimensional inverse scattering problems. Russ. Math. Surv.42, 109–180 (1987) · Zbl 0674.35085 · doi:10.1070/RM1987v042n03ABEH001419
[11]	[P] Prosser, R. J.: Formal solutions of inverse scattering problems I, II, III. J. Math. Phys.10, 1819–1822 (1969); J. Math. Phys.17, 1225–1779 (1976); J. Math. Phys.21, 2648–2653 (1980) · doi:10.1063/1.1664766
[12]	[W] Weder, R.: Generalized limiting absorption method and multidimensional inverse scattering theory. preprint (1990) · Zbl 0738.35055