Scattering on a line. I. Existence of wave operators and asymptotic completeness in the periodic case

Existence of wave operators for scattering on a line in R³ is proved. It is shown that the completeness of have operators is violated in the case of a line with periodic boundary conditions. However, the asymptotic completeness and unitary property of wave operators are preserved. Bibliography:11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 136–141, 1990. Translated by Ya. V. Kurylev.