= Solution
The <quantum propagator> is the position-space kernel of the <time-evolution operator>: $\Psi(x,t)=\int K(x,t|x',t')\Psi(x',t')dx'$. For a time-independent <Quantum Hamiltonian>, $U=e^{-iH(t-t')/\hbar}$. Inserting a complete orthonormal <energy> eigenbasis gives
$$
\boxed{K(x,t|x',t')=\sum_\alpha\varphi_\alpha(x)\varphi_\alpha^*(x')e^{-iE_\alpha(t-t')/\hbar}}.
$$
A <continuous spectrum> requires the corresponding <integral>. At equal times, completeness makes this kernel a <Dirac delta function>.
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