Consider the energy conservation principle, than the total amount of energy in the universe is a fixed value $E$.
Let us denote with $| \psi \rangle$ the wave function of the entire universe.
Is it right to write the Schödinger equation for the entire universe as
$ i \hbar \frac{d |\psi \rangle}{dt}=E |\psi \rangle$ ?
If this is right, it means that the universe's wave function is fundamentally stable?
Well, why don't we consider a smaller problem before we tackle the whole universe. For instance, a particle with a wavefunction $\psi(t)$ in an infinite potential well that has eigenstates with energies $E_0$, $E_1$, $E_2$, etc...
This particle can always be written as a superposition of energy states.
$$ |\psi(t) \rangle = \sum_{i=0}^\infty c_i e^{iE_it} |E_i \rangle $$
Now, what is the energy of the particle? From this equation, the particle can exist in many possible different total energy states based on some probability. Thus, even if you have steady state solutions, that does not guarantee you know the exact total energy of the system. One can instead look at the average total energy
$$ \langle E \rangle = \langle \psi(t) | H | \psi(t) \rangle $$
but there is no guarantee that $\langle E \rangle$ is equal to any eigenenergy $E_i$. So, if we consider your E the average energy of the universe, this does not mean we are in a steady state.
