Spectral resolution of a quantum propagator ($K_U(x,t;x^{\prime },t^{\prime })={\sum }_{\alpha }{\phi }_{\alpha }(x){\phi }_{\alpha }^{\ast }(x^{\prime })e^{-i{E}_{\alpha }(t-t^{\prime })/\hslash }$)

Insert a complete resolution of identity in energy eigenstates on either side of the time-evolution operator. Each state evolves by its scalar energy phase, yielding the displayed quantum propagator. Its spectral sum includes an integral over any continuous spectrum, with the measure fixed by the continuum normalization. Omitting the scattering states when bound states are also present destroys completeness and the delta-function short-time limit.