= Solution
Expand the <wavefunction> in the complete orthonormal <energy eigenstates>. The <Time-dependent Schrödinger equation> makes each coefficient satisfy $i\hbar\dot a_n(t)=E_na_n(t)$, so
$$
\boxed{\psi(x,t)=\sum_{n=1}^{\infty}a_n e^{-iE_nt/\hbar}\psi_n(x).}
$$
The series is interpreted in the <Hilbert space> norm. Its norm is conserved because the phase factors have modulus one and the <energy eigenstates> are orthonormal. For initial data in the domain of the <Hamiltonian operator>, this is a strong solution of the equation; for any square-summable coefficients it gives the corresponding <unitary time evolution>.
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