 Solution

At the origin, the normalized bound state in a delta potential contributes $|{\phi }_0(0){|}^2=\kappa$, and the odd scattering eigenfunctions contribute zero. The even continuum functions from the preceding part haveTheir energy is $E_k={\hslash }^2{k}^2/(2m)$. Inserting these weights into the spectral resolution of a quantum propagator, and using evenness to extend the integral to the whole real axis, yields for $T>0$This is the quantum propagator of an attractive delta potential. The printed continuum exponent has an extra factor of $\hslash$: its time phase must be $-i{E}_{k}T/\hslash =-i\hslash k^{2}T/(2m)$. This correction also makes the exponent dimensionless. Oscillatory integrals are understood with an infinitesimal positive convergence factor before taking its limit.