Let's say that I am inhave some quantum system defined by Hamiltonian $\hat{H}$. The energy eigenstates of this Hamiltonian form a complete basis for the Hilbert space of all possible states corresponding to this quantum system.
Now, if I have some initial state $|\Psi(0)\rangle$, I can solve for $|\Psi(t)\rangle$ by finding the complex coefficients in the expansion of $|\Psi(0)\rangle$ in the energy basis, then add the usual time-dependence and append each of the coefficients to their corresponding eigenstate in a linear combination.
My question is, what if $|\Psi(0)\rangle$ is an unphysical state of our quantum system (does not obey the Schrödinger equation for this particular Hamiltonian)? Am I allowed to choose this as my initial state? If so, how do I know I can expand it in terms of energy eigenstates corresponding, as to my understanding, these form a complete basis for the Hilbert space of possible states for this Hamiltonian, when $|\Psi(0)\rangle$ is an unphysical state that lies outside the Hilbert space (does it lie outside the Hilbert spacea given system?).
Even more generally, is the Hilbert space of states for a given system restricted by the Hamiltonian itself?