 Solution

Away from $x=x^{\prime }$, the free equation is $G^{\prime \prime }+k^{2}G=0$. The outgoing boundary condition selects $G=A{e}^{ik|x-x^{\prime }|}$, continuous at its source. Integrating the differential equation across the source gives the derivative jumpThe jump for the proposed form is $2ikA$, so $A=im/(k{\hslash }^2)$. ThusFor $x$ far to either side, the $x^{\prime }$-dependent phase is absorbed into the outgoing amplitude. The outgoing prescription is equivalently the boundary value of the resolvent with $E\to E+i0$; changing that prescription would give a different Green function.