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