Abstract
The Schrödinger equation with a time-dependent Hamiltonian is considered in the space . It is assumed that , , , and ; . 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 there is a sequence such that , then, for some , , . This provides an effective description of the ranges of the wave operators relating the problems with the free Hamiltonian and the complete Hamiltonian . Examples show that the conditions imposed are best possible. The case of functions periodic in is treated separately; in this case the description of the ranges of the wave operators can be given in spectral terms for and any . More general differential operators are also considered. Bibliography: 14 titles.
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