Inverse Scattering Problem For The Schr?dinger Equation With A Symmetric Potential

Historically interest in inverse scattering theory began with the difficulties of theoretical physicists in trying to construct a rigorous quantum field theory.Since the breakthrough of one-dimensional inverse scattering theory achieved by Gal'fand and Levitan in 1950's,Krein,Marchenko and Faddeev did extensive investigations.In this paper,we discuss the inverse scattering problem for the Schr?dingerequation with a symmetric potential in high dimensions,the function in L2(Rn)is regarded as a product of a function that depends only on the distance from the origin and a function on the sphere,using the space decomposition,we get a way to decompose the wave operator W±in high dimensions into the tensor product of the wave operator Wj±in one dimension and the unit operator.None the less erenow,the paper of P.Deift and E.Trubowitz showed the result of potential V(x)? L12(R)in one dimension in 1979,which we can use to get a result in high dimensions in this paper.The main proof train of thought of space decomposition comes from the articles of Simon and Hormander.