×

Inverse scattering problems in several dimensions. (English) Zbl 0820.35138

The Schrödinger equation \((- \Delta + q(x) - k^ 2)u = 0\) is considered, where \(u(x)\) is the sum of incident plane wave \(e^{ik \omega \cdot x}\) and the reflected wave \(v(x, \omega,k)\), the latter having the asymptotics \[ v(x, \omega,k) = {e^{ik | x |} \over | x |^{{n-1 \over 2}}} \left( a (\theta, \omega,k) + O \left( {1 \over | x |} \right) \right), \] which leads to an overdetermined problem. Restrictions based on previous works of the authors [Commun. Math. Phys. 124, 169-215 (1989; Zbl 0706.35136) and ibid. 138, 451-486 (1991; Zbl 0728.35146)] are derived to use the backscattering amplitude \(a(\theta, - \theta,k)\) for the determination of real-valued \(q(x)\). Maps of the backscattering amplitude \(S\) are discussed. The one-dimensional case and the case with \(n \geq 3\) are treated. The case with \(n = 2\) needs a special consideration due to a singularity.
Reviewer: V.Burjan (Praha)

MSC:

35R30 Inverse problems for PDEs
35P25 Scattering theory for PDEs
81U40 Inverse scattering problems in quantum theory