Now the eigenfunctions of the Hamiltonian clearly differ from one problem to another since the Hamiltonian depends on the potential and hence for a different potential we get a different eigenvalue equation for the Hamiltonian hence the eigenfunctions are different each time.
However, the eigenvalue equation for the position and momentum operators don't change (since they don't depends on the potential), so is it always the case that the eigenfunctions of momentum and position operators (which are continuous) are always the same? My gut tells me that no since the eigenfunctions must also satisfy boundary conditions which differ from one problem to another, but Griffiths book only solved this one case for the eigenfunctions of position and momentum operators and later used them as a standard eigenbasis of position/momentum for all types of problems. Even in the end of chapter problems he never asks us to solve the eigenvalue equation for the position and momentum operators for, say, the infinite well. Which left me with the aformentioned question: Are they always the same?