Scattering subspaces and asymptotic completeness for the time-dependent Schr\"odinger equation

D. R. Yafaev

Sbornik. Mathematics, Tome 46 (1983) no. 2, pp. 267-283

Résumé

The Schrödinger equation $i\partial u/\partial t=H(t)u$ with a time-dependent Hamiltonian $H(t)=-\Delta+q(x,t)$ is considered in the space $L_2(\mathbf R^m)$. It is assumed that $q=\overline q$, $|q(x,t)|\leqslant c(1+|x|)^{-a}$, $a>2$, and $m\geqslant5$; $H_0=-\Delta$. It is shown that each solution of the Schrödinger equation which exits any compact subset of configuration space must have free asymptotics. More precisely, if for any $\rho$ there is a sequence $~t_n\to\pm\infty$ such that $\int_{|x| \rho}|u(x,t_n)|^2\,dx\to0$, then, for some $f_\pm$, $\|u(t)-\exp(-iH_0t)f_\pm\|\to 0$, $t\to\pm\infty$. This provides an effective description of the ranges of the wave operators relating the problems with the free Hamiltonian $H_0$ and the complete Hamiltonian $H(t)$. Examples show that the conditions imposed are best possible. The case of functions $q(x,t)$ periodic in $t$ is treated separately; in this case the description of the ranges of the wave operators can be given in spectral terms for $a>1$ and any $m$. More general differential operators are also considered. Bibliography: 14 titles.