Multidimensional Inverse Scattering with Fixed-Energy Data

1 Introduction In this lecture the author reviews his results on multidimensional inverse scattering. References to the works of other authors can be found in [20]. Five topics are briefly discussed:-1) property C with constraints and new type of the uniqueness theorems for inverse scattering,-2) inversion of noisy discrete fixed-energy 3D scattering data and error estimates,-3) variational principles equivalent to inverse scattering problems,-4) low-frequency data inversion,-5) asymptotic inverse scattering theory for scattering by small inhomo-geneities. Detailed proofs of the results can be found in the cited references. In this paper the emphasis is on the ideas and formulation of the results. The inverse problems we consider are: inverse potential, geophysical and obstacle scattering (IPS, IGS, IOS). We recall the statements of these problems. Let [V 2+k 2-q(x)] u=0inlR 3, k= const >0 (I.1) k)exp(ikr) (1) u = exp(ika, x) + A(a I,a, ~ r + o , X .9 Y where S 2 is the unit sphere, c~ is a given unit vector. A(a ~, a, k) is the scattering amplitude, and q(x) is the potential. We assume that q • Q := {q : q = g (real-valuedness), q(x) = 0 for Ixl >_ a, q • L~}, k > 0 is fixed. The solution to (1.1)-(1.2) is called the scattering solution. We denote Ba the ball centered at the origin with radius a, u0 := exp(ikc~, x), Mk := {z : z • e a, z. z = k2}, 3 z-y := }-'~j=l zjyj, M1 := M, Mk is a non-compact algebraic variety in C a, M V/lR a = S 2 .