Poles in the Lippmann-Schwinger Equation

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  • #1
While deriving the Helmholtz Green function in Sakurai we come across the integral
[tex]\int_{-\infty}^{\infty}q\,dq\,\frac{e^{iq|\vec x-\vec x'|}-e^{-iq|\vec x-\vec x'|}}{q^2-k^2\mp i\varepsilon'}[/tex]

This equation has poles at [tex]q \simeq \pm k\pm i\varepsilon'[/tex], however when doing the residue calculation it seems that Sakurai only treats the poles [tex]k+i\varepsilon'[/tex] and [tex]k-i\varepsilon'[/tex], but not the companion poles poles [tex]-k-i\varepsilon'[/tex] and [tex]-k+i\varepsilon'[/tex].

Is there a physical reason for this I am missing or do I have a mathematical error? If included, it seems the other poles would give both the [tex]\psi^{(\pm)}[/tex] solutions over again?

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  • #2
It looks like he's considering all four poles actually. When you offset the poles, to [itex]k+i\varepsilon'[/itex] and [itex]-k-i\varepsilon'[/itex], where [itex]\varepsilon'>0[/itex], one pole is included in contour for the [itex]e^{iq|\vec{x}-\vec{x}'|}[/itex] integral, and the other pole is inside the contour for the [itex]e^{-iq|\vec{x}-\vec{x}'|}[/itex] integral. Both integrals contribute to the [itex]e^{+ik|\vec{x}-\vec{x}'|}[/itex] solution. The other two poles correspond to [itex]\varepsilon'<0[/itex] and yield the [itex]e^{-ik|\vec{x}-\vec{x}'|}[/itex] solution. One is an incoming wave; the other is an outgoing wave. The choice of how you move the poles depends on your boundary conditions. Arfken (3rd edition), if you have it, discusses the integration on page 919.

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