zbMATH — the first resource for mathematics

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

35J10 Schrödinger operator, Schrödinger equation
35P25 Scattering theory for PDEs
Full Text: Numdam EuDML