Quasi-classical asymptotics of the section of scattering for asymptotically homogeneous potentials. (Russian. English summary) Zbl 0621.35074

Sobolev, A. V.

Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 152, 105-136 (1986).

The total scattering cross-section by a potential gV(x), \(x\in {\mathbb{R}}^ m\), \(m>3\), is considered for large coupling constants g and wave numbers K. It is supposed that \(V(x)\sim \Phi (x/| x|)| x|^{- d}\), \(2\alpha >m+1\), as \(| x| \to \infty\). It is shown that as \(gk^{-1}\to \infty\), \(g^{3-\alpha}k^{2(\alpha -2)}\to \infty\) the cross-section asymptotically equals \(\theta_{\alpha}(gk^{- 1})^{\kappa}\), \(\kappa =(m-1)(\alpha -1)^{-1}\). Here the coefficient \(\theta_{\alpha}\) is determined only by the function \(\Phi\) and the number \(\alpha\). Under additional assumptions \(\Phi >0\), \(V>0\) this asymptotics holds in the broader region \(gk^{-1}\to \infty\), \(gk^{\alpha -2}\geq c(gk^{-1})^{\delta}\), \(\delta >0\).

MSC:

35P25	Scattering theory for PDEs
35Q99	Partial differential equations of mathematical physics and other areas of application

Keywords:

quasi-classical asymptotics; total scattering cross-section; potential; large coupling constants