Reading Sam Treiman's odd quantum he nicely explains the dependencies between the Schrödinger wave equation, eigenvalues and eigenfunctions (page 86 onwards). In his notation, eigenfunctions are [itex]u:R^3\to R[/itex] and the wavefunction is [itex]\Psi:R^4\to R[/itex], i.e. in contrast to the eigenfunctions it depends on time.

Then on page 94 he says:

With "state of the system" he refers of course to [itex]\Psi[/itex], so during the measurement, the jump or collapse is from [itex]\Psi[/itex] to [itex]u[/itex].

The one thing I don't understand here is: [itex]u[/itex] does not depend on time, so how is the development of the new [itex]\Psi[/itex] over time governed? Is it that every solution of the Schrödinger equation is uniquely determined as soon as the value at just one point in time is known?