Can the Lippmann-Schwinger equation be integrated over negative r values?

[Manojg]

Hi,

I have a simple question.
I am looking at Sakurai's "Modern Quantum Physics, Revised edition" on page 382 where he tries to integrate the Lippmann-Schwinger equation. From equation 7.1.15 to 7.1.16, he converted from Cartesian to spherical coordinate system. After integration over $$\phi$$ and $$cos\theta$$, he changed the integration over "q" (which is radius in spherical system) from "0 to +infinity" to "-infinity to +infinity".

One can't change radius from -infinity to +infinity in spherical coordinate, right? Then, how did he get that equation?

Thanks.

 

[Bill_K]

I don't have the book, but sure you can do that if the integrand is even. Doesn't mean there is anything physical about negative r, it's just a formal way of evaluating the integral. Maybe what comes next is a contour integration in the complex r plane?