The phase analysis in the problem of scattering on a radial potential. (Russian. English summary) Zbl 0588.47011

Sobolev, A. V.; Yafaev, D. R.

Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 147, 155-178 (1985).

The behaviour of the total scattering cross section averaged over all directions \(\sigma\) (k,g) was studied, first at g/k\(\to \infty\), of a three-dimensional quantum particle of energy \(k^ 2\) with a radial potential gV(r), g is the binding constant. Assuming \(V(r)\sim V_ 0r^{-\alpha}\), \(\alpha >2\), \(r\to \infty\), the asymptotics \(\sigma (k,g)\sim \kappa_{\alpha}(| V_ 0| g/k)^{2\lambda \alpha}\), \(\lambda_{\alpha}=(\alpha -1)^{-1}\) was obtained in the range \(g^{3-\alpha}k^{2(\alpha -2)}\to \infty\). The coefficient \(\kappa_{\alpha}\) was expressed explicitly using the \(\Gamma\)-function. A proof was given. It is based on the phase functions in the potential scattering [F. Kolodzhero, Method of phase functions and theory of potential scattering (Russian) (1972)]. At \(V\geq 0\) the asymptotics is valid even in the broader range g/k\(\to \infty\), \(gk^{\alpha -2}\to \infty\). Then the asymptotics g/k\(\to 0\) (high energies), with potentials gV(r), \(g\geq 0\), \(r\to 0\) was derived to be \(\sigma (k,g)\sim \kappa_{\beta}(V_ 0g/k)^{2\lambda \beta}\), \(gk^{\beta -2}\to \infty\), at \(V(r)\sim V_ 0r^{-\beta}\), \(V_ 0>0\), \(\beta >2\). The asymptotics of the forward scattering amplitude was calculated similarly.

Reviewer: V.Burjan

MSC:

47A40	Scattering theory of linear operators
35P20	Asymptotic distributions of eigenvalues in context of PDEs
81U99	Quantum scattering theory

Keywords:

total scattering cross section; three-dimensional quantum particle; binding constant; asymptotics; phase functions in the potential scattering; forward scattering amplitude