Scattering problem for radial Schr\" odinger equation with a slowly decreasing potential

V. B. Matveev ; M. M. Skriganov

Teoretičeskaâ i matematičeskaâ fizika, Tome 10 (1972) no. 2, pp. 238-248

Résumé

The scattering problem for the Schrodinger equation with slowly decreasing potential is considered. Stationary wave operators $W_{\pm}(H,H_0)$ are constructed and their completeness is proved. It is shown that the operators $W_{\pm}(H,H_0)$ can also be defined as the limits $W_{\pm}(H,H_0)=\lim{t\to\pm\infty} \exp(itH)T_{\pm}\exp(-itH_0)$, $T_{\pm}$ being some operators, which do not depend on $t$, do not commute with $H_0$ and can be constructed explicity for the :given potential $q(x)$.The invariance principle for the wave operators $W_{\pm}$ is proved.