×

Properties of the scattering map. (English) Zbl 0595.34027

Let us consider the scattering map associated with the Schrödinger equation \(-y''(x)+q(x)y(x)=k^ 2y(x)\) on the whole real line, defined as follows: Denote by \(f_ i(x,k)=f_ i(x,k,q)\) \((i=1,2)\) the solutions of this equation for Im \(k\geq 0\) with \(f_ 1(x,k)\sim e^{ikx}\) (x\(\to \infty)\) and \(f_ 2(x,k)\sim e^{-ikx}\) (x\(\to -\infty)\) respectively. The Wronskian \([f_ 1(x,-k),f_ 2(x,k)]\), considered as a function of q, is called the scattering map. The authors show among other things that in appropriate spaces the scattering map is a real analytic isomorphism when restricted to the set of potentials q without bound states.

MSC:

34L99 Ordinary differential operators
PDFBibTeX XMLCite
Full Text: DOI EuDML