We know there are some (many actually) exactly solvable models, like the Hydrogen atom, the harmonic oscillator, etc. But these models are solvable often only in the sense that the eigenstates or eigenvalues are solvable. In other words, the corresponding time-independent Schroedinger equation is exactly sovable.
How about the time-dependent Schroedinger equation? How many models are exactly solvable in the sense of its dynamics?