= Quantum propagator of an attractive delta potential
{title2=$K(0,T;0,0)=\kappa e^{i\hbar\kappa^2T/(2m)}+\int_{\mathbb R}\frac{dk}{2\pi}\frac{k^2e^{-i\hbar k^2T/(2m)}}{k^2+\kappa^2}$}
For $H=p^2/(2m)-g\delta(x)$ with $g>0$, put $\kappa=mg/\hbar^2$ and $T>0$. The <bound state in a delta potential> has squared amplitude $\kappa$ at the origin. The even normalized continuum states have squared amplitude $k^2/[\pi(k^2+\kappa^2)]$ for $k>0$, and the odd states vanish there. The <spectral resolution of a quantum propagator> gives the displayed result. At long times, the continuum stationary point at $k=0$ is suppressed by its quadratic spectral weight: expanding that weight gives a continuum term $e^{-3i\pi/4}(2m/\hbar T)^{3/2}/(4\sqrt\pi\kappa^2)+O(T^{-5/2})$. The <bound state> term persists with constant modulus. This <stationary phase> expansion requires $\hbar\kappa^2T/(2m)\gg1$.
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