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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.

For fixed $g>0$, write $A=\hslash T/(2m)$. The bound contribution to the quantum propagator of an attractive delta potential remains $\kappa e^{i\hslash {\kappa }^{2}T/(2m)}$. For its continuum, the stationary phase is at $k=0$, where $k^2/(k^2+{\kappa }^2)=k^2/{\kappa }^2+O(k^4)$. The Gaussian oscillatory integral and its derivative giveConsequentlyThe continuum decays as $T^{-3/2}$, but the full kernel has a persistent oscillating bound state contribution and no ordinary complex limit. This expansion requires $\hslash {\kappa }^{2}T/(2m)\gg 1$; it is not uniform in the zero-coupling limit, where the bound state disappears and the free kernel instead decays as $T^{-1/2}$.