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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.

A pole of the continued scattering matrix associated with outgoing boundary conditions and an unstable trapped state. A pole below the real energy axis gives time dependence $e^{-i{E}_{R}t/\hslash }{e}^{-\Gamma t/(2\hslash )}$, so the probability lifetime is $\hslash /\Gamma$. Bound-state poles instead lie on the positive imaginary wavenumber axis and produce square-integrable decaying spatial waves.

A scattering state describes an incoming preparation and its outgoing components after an interaction. In a stationary calculation, it is typically a generalized energy eigenstate with specified incoming boundary data, rather than a square-integrable bound state.

For a short-range potential, its distant form separates freely propagating incoming and scattered waves. The outgoing boundary condition selects how a localized disturbance radiates away, and the resulting relative coefficient determines a scattering amplitude. A physical normalizable packet is formed by superposing stationary scattering states over an energy or momentum range. This distinction allows finite particle probabilities and localized preparations without treating an infinite plane wave as a normalized state inside the physical Hilbert space.