Hallo I'm trying to understand the concept of representation in the position space. I read that $|x\rangle$ are the eigenstates of the position operator, but I think this states should evolve in time since there aren't stationary states with a precise position?
What does $|x\rangle$ really mean?
The states $|x\rangle$ are eigenstates of the position operator, and they do not change with time.
That means that they are not solutions of the Schrödinger equation. This is fine: not every state in the Hilbert space* needs to evolve with time or obey the Schrödinger equation.
If you do take a position eigenstate as the initial state for a Schrödinger-equation evolution, then the state will obviously evolve, since it is not in an eigenstate of the hamiltonian. For a free particle, it will immediately spread over all of space; for the details, see The Dirac-delta function as an initial state for the quantum free particle.
*That's a minor cheat - the position eigenstates are not actually in the Hilbert space. That doesn't affect the conclusions, though.
