"Ruling out" eigenvalues in a superposition It recently occurred to me that my knowledge of quantum mechanics doesn't really include anything about measurements that don't return definitive answers.
Some examples might include a system in superposition with 3 eigenstates and a measurement is performed with a negative result (negative meaning it wasn't found), such as looking for the particular eigenvalue associated with eigenstate |1> and not finding it (or maybe a more intuitive example would be a position wavefunction, and making a measurement within only a certain region and not finding the particle there).
What I'm wondering about specifically is what can be said, if anything, about the remaining states? We have something like the "Quantum Zeno Effect" for measurements with positive results, but is there anything similar for measurements with negative results?
For instance, if I measure a region and find a particle not to be there, will very rapid successive measurements always return negative results and prevent me from ever observing a particle there? Or similarly, does obtaining a negative result in region A and then performing a very rapid successive measurement on region B increase my odds of finding the particle in region B, effectively meaning the probability elsewhere increases due to region A having been "ruled out" and the fact that the total probability must still equal 1?
 A: There is no such thing as performing a measurement with a negative result. When performing a measurement, the wave function collapses by definition into one of the eigenstates it is expanded onto.
In your example 

such as looking for the particular eigenvalue associated with eigenstate |1> and not finding it

that is not a measurement: a measurement would be looking for an eigenvalue of operator $\hat{A}$, whatever it is, and writing it down. This means that if you do not find the final state to be (in your example) the eigenstate $|1\rangle$, the wave function will nevertheless be in any other eigenstate, say $|n\rangle$ and that would still be a positive outcome of a measurement process.
Your second example is trickier because the position does not have eigenstates, namely every wave function is always everywhere spread in the universe and cannot be localised; however, if we still start from the hypothesis that there are such eigenstates and that the wave function can be expanded onto such a base, by definition, for the eigenfunction to be in the domain of the observables (in this case the position), then at least one position outcome of the basis must be in the domain you are performing the measurement in, therefore even in this case there is no scenario of not finding the collapsed state anywhere: for this description to be consistent, there must be at least one eigenstate in your domain onto which the the wave function may collapse.
