Asymptotic completeness in the problem of scattering by a~Brownian particle

S. E. Cheremshantsev

Sbornik. Mathematics, Tome 65 (1990) no. 2, pp. 531-559

Résumé

The author studies the three-dimensional Schrödinger equation with potential randomly depending on time: $$ i\frac{\partial\psi}{\partial t}=-\Delta_x\psi+q(x-y(t))\psi;\quad\psi|_{t=0}=\psi_0(x);\quad t\geqslant0. $$ Here $\psi_0\in L_2(\mathbf R^3)$, $q$ is a fixed complex function, $y(t)$ is a sample function of the Wiener process. The main result is the following. Let $\operatorname{Im}q(x)\leqslant0$, $q\in L_2(\mathbf R^3)$ and suppose there exist $R$, $\delta>0$, such that $|q(x)|\leqslant C|x|^{-7/2-\delta}$ for $|x|\geqslant R$. Then for almost all (relative to Wiener measure) $y(\,\cdot\,)$ the solution $\psi(t,y(\,\cdot\,))$ of the above equation has free asymptotics as $t\to+\infty$ for any initial data $\psi_0$ in $L_2(\mathbf R^3)$, i.e. for some $\psi_+$ $$ \lim_{t\to+\infty}\|\psi(t,y(\,\cdot\,))-\exp(-itH_0)\psi_+\|_{L_2(\mathbf R^3)}=0,\qquad H_0=-\Delta_x. $$ Bibliography: 13 titles.