In a course that I follow, we use the perturbation method to find the eigenvectors and energies to an Hamiltonian written $ H_0 + V $ where $V$ is a weak perturbation.
It is written that as $V$ is a weak perturbation, we can write the perturbed eigenvectors as a combination of the eigenvectors of the unperturbed hamiltonian.
But I thought that as we know the eigenvectors of the unperturbed Hamiltonian, then we have a basis of the Hilbert space so there is no need of this assumption (we always can write any vector as a linear combination of eigenvectors of the unperturbed hamiltonian).
Can you help me to understand this?
