In my quantum mechanics lecture notes(see picture at the bottom), they say the plane wave basis $\{\phi_k(x)=\frac{1}{(2\pi)^{3/2}}\exp(ik\cdot x)\}$ is so general that any $ \psi \in L^2(\mathbb{R}^3)$ can be expressed in that basis, as equation $(34)$ shows and then they write in equation $(35)$ but highlighting that this is for the special case of the S.E. for the free-particle.
My question is, why is this expresion not valid in general? I don't think I need a free particle to do it, because the S.E always has the time-dependent part solved by $$e^{-\frac{iEt}{\hbar}}$$ and we only deal with the eigenvalue equation for the hamiltonian to get the space-dependent part $\psi(x)$ and costruct the whole solution as $$\psi(x,t)=\psi(x)e^{-\frac{iEt}{\hbar}}$$ In the case of the free partiche the solution of the eigenvalue equation of the hamiltonian is $$\psi(x)=\phi_k(x) = e^{ik.x}$$. If the particle was not free the complete solution would still be $$\psi(x,t)=\psi(x)e^{-\frac{iEt}{\hbar}}$$ with $\psi(x)$ the solution of the corresponding eigenvalue problem and the equation (35) should be true in general