Wave operators for the Schr\"odinger equation with a~slowly decreasing potential

V. S. Buslaev ; V. B. Matveev

Teoretičeskaâ i matematičeskaâ fizika, Tome 2 (1970) no. 3, pp. 367-376

Résumé

The present article is devoted to the study in space $L_2(R^n)$ of the energy operator $\displaystyle H_q=-\frac 1{2m}\Delta+q(x)$, where the function $q(x)$ decreases slower that $|x|^{-\alpha}$, $\alpha>0$, as $|x|\to\infty$. An explicit “regularizing” operator $U_q(t)$ is constructed and the existence of generalized wave operators $$ W_{\pm}(H_q, H_0)=\mathop{\textrm{s-lim}}_{t\to\pm\infty}\exp\{-itH_q\}\exp\{itH_0\}U_q(t) $$ is proved.