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The Lippmann-Schwinger equation treated as a characteristic Cauchy problem. (English) Zbl 0702.35065
Sémin. Équations Dériv. Partielles 1988-1989, Exp. No. 4, 7 p. (1989).
The author considers the standard solution of the Lippmann-Schwinger equation $$(-\Delta_ x+v(x))\phi (x,\theta,k)=k^ 2\phi (x,\theta,k)$$, where $$k^ 2$$ is the energy, $$\theta$$ is an angular variable, and v in a class $${\mathcal V}$$ of short range potentials. It is proved that $e^{- ik<x,\theta >}\phi (x,\theta,k)=1+\int^{\infty}_{0}w_{\theta}(x,t)e^{ikt}dt,$ where $$w_{\theta}$$ is smooth. The proof relies on some intertwining technique combined with the examination of a characteristic Cauchy problem for the operator $${\mathcal L}_{\theta}=\Delta_ x-\partial_ t^ 2-v(x)$$. Since $${\mathcal L}_{\theta}(\delta (t)+Y(t)w_{\theta}(x,t))=0$$, where Y is the Heaviside function, there is a simple recursion formula for the derivatives of $$w_{\theta}(x,t)$$ when $$t=0$$. In particular one obtains the equation $$2<\theta,\partial_ x>w_{\theta}(x,0)=v(x)$$. This relation was discovered by R. G. Newton, and it was called ‘the miracle’ by him.
Reviewer: A.Melin

##### MSC:
 35J10 Schrödinger operator, Schrödinger equation 35P25 Scattering theory for PDEs
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