I'm probably missing something obvious and basic here but I can't make sense of certain usages of Observables as present in basic treatments of Quantum Mechanics that i've come across.
$$ \hat{A}|\Psi\rangle = a|\Psi\rangle $$
The above equation implies to me that a single eigenket gives a single eigenvalue of $\hat{A}$.
However Ket Vectors that are composed of superpositions have multiple possible eigenvalues. Which leads me to believe that that equation is only valid for eigenkets which are Basis States.
However in the Schrödinger equation we have an Observable (Hamiltonian) acting on Wave Functions in Position Space which are composed of an infinite number of Basis States.
In this usage is it somehow assumed that every Basis State in the Position Basis corresponds to a single Energy Eigenstate? (I wouldn't think this would be the case. But what is the point/result of applying the Hamiltonian to any given Wave Function then?)
Further confusion arises from this because if the Energy is exactly known then shouldn't there be some sort of maximal uncertainty in time?
As a final question is there any kind of useful interpretation of multiplying the eigenket by it's eigenvalue as appears in the above Observable Equation? In all treatments I've seen this multiplication is simply ignored and the eigenvalue itself is the only focus.