I'm continuing this question because the answer to this is not helping me. As the OP said the propagator in the problem should be given in energy basis :
$$U(t)=\sum_{\alpha=\pm}\int_0^\infty |E,\alpha\rangle\langle E,\alpha | e^{-iEt/\hbar} dE$$
But if you derive it from the expression of momentum basis : $$U(t)=\int_{-\infty}^\infty |p\rangle\langle p|e^{-i E(p)t/\hbar}dp$$ by substituting the value of $E=E(p)$ then you get an additional factor for density of state.
I'm asking for a solution that starts from the energy basis such that the density of the state came out automatically. As the propagator is just $$U(t)=e^{-iHt/\hbar}$$
