Solution of time-dependent Schrödinger equation in a nontraditional Hilbert space

A Hilbert space that is not traditional for quantum mechanics is used. It is the space of square-integrable functions of not only the space and spin variables $\mathbf r$ but also the time $t$ on an interval $(-T/2, T/2)$. In this space, the time-dependent Schrödinger equation formally reduces to a time-independent equation, which can be solved by the methods of time-independent scattering theory. A solution is found for cases in which a time-dependent perturbation contains several harmonics; this perturbation is small, and the duration of the perturbation is appreciably shorter than the lifetime of the excited levels.