= Solution
Choose the even-channel <scattering phase shift> convention $\varphi_{k,e}(x)=\pi^{-1/2}\cos(k|x|+\delta_e)$ for $k>0$. The <delta potential> derivative jump gives
$$
-2k\sin\delta_e=-2\kappa\cos\delta_e,
$$
so, choosing the branch that tends to zero as $k\to\infty$,
$$
\boxed{\delta_e(k)=\arctan(\kappa/k),\qquad \delta_o(k)=0.}
$$
An odd <scattering eigenfunction> is $\varphi_{k,o}(x)=\pi^{-1/2}\sin kx$: it vanishes at the interaction and its derivative is continuous, so it acquires no <scattering phase shift>. These parity channels, with the <bound state>, give the continuum part of the <spectral resolution of a quantum propagator>.
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