Inverse scattering on half-line
The author gives a new approach to the theory. This approach uses a Wiener-Hopf type equation (which is Marchenko's equation at \(x=0)\). The normalizing constants are expressed in terms of the parameters which define the general solution to this equation. A detailed analysis of invertibility of the steps in the various inversion procedures is given. In particular several ways are given to prove that the potential reconstructed from the scattering data generates the original scattering data; that the scattering data, corresponding to a potential, generate via the inversion procedure the original potential. Homeomorphism between a class of potentials and a class of scattering data is established. The class of potentials q is \(\{\) \(q: q=\bar q,\int ^{\infty}_{0} (1+x)| q(x)| dx<\infty \}\). Applications are given. In particular, an example of \(q\in C_ 0^{\infty}(0,R)\) is given for which there exist infinitely many purely imaginary resonances; a necessary and sufficient condition on the scattering data is given for \(q\in L^ 2(0,\infty)\), and some other results are obtained.
scientific article; zbMATH DE number 4026012
scientific article