Conservation of Probability

The Schroedinger equation is a recipe for calculating |Ψ(t)> when |Ψ(t0)> is known.  The Schroedinger equation is linear.  The correspondence between |Ψ(t)> and |Ψ(t0)> is therefore linear. 
There exists a linear operator that transforms |Ψ(t)> into |Ψ(t0)>.
|Ψ(t)> = U(t,t0) |Ψ(t0)>.

What are the properties of this evolution operator?

Let |En> be an eigenstate of H and let |Ψ(t0)> = |En>.  Then
|Ψ(t)> = exp(-iH(t - t0)/ħ)|En> = exp(-iEn(t - t0)/ħ)|En>
is also an eigenstate. If H does not explicitly depend on time and the system is in an eigenstate of H at t = t0, then it will remain in this eigenstate.  The system is in a stationary state.