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The eikonal approximation and the asymptotics of the total scattering cross-section for the Schrödinger equation. (English) Zbl 0608.35054
Let $$\sigma(\omega,k,v)$$ be the total cross section in the scattering problem for the Schrödinger equation $$-\Delta \psi +v(x)\psi =k^ 2\psi$$ in $${\mathbb{R}}^ m$$, $$m\geq 2$$, where $$k>0$$, $$\omega \in S^{m-1}$$, $$\omega$$ k is the incident momentum. The author assumes that $$v(x)=gq(x)$$, where $$g\in {\mathbb{R}}$$ and $$q(x)=r^{-\alpha}\Phi (\hat x)(1+o(1))$$, $$r\to \infty$$, $$r=| x|$$, $$\hat x=xr^{-1}$$, $$2\alpha >m+1$$, $$\Phi(\omega)\in C(S^{m-1})$$. Besides q(x) is twice differentiable with respect to r and $$\sup_{x\in {\mathbb{R}}^ m} (| q(x)| +r| \partial q/\partial r| +r^ 2| \partial^ 2q/\partial r^ 2|)<\infty.$$
If $$g\to \infty$$, $$k\to \infty$$, $$N=g(2k)^{-1}\to \infty$$, but $$g\leq \gamma_ 0k^ 2$$ with $$\gamma_ 0\leq [\inf_{\beta \in (0,2)} \sup_{x\in {\mathbb{R}}^ m} (\beta^{-1}r(\partial q/\partial r)+q)]^ 1$$, then $\sigma (\omega,k,gq)=\pi [(m-1)\Gamma (\kappa)\sin (\pi \kappa /2)]^{-1} N^{\kappa}\int_{S_{\omega}^{m-2}}| \Omega (\omega,\phi)|^{\kappa}def(1+o(1))$ where $$\kappa =m-1/\alpha -1$$ $\Omega (\omega,\phi)=\int^{\pi}_{0}\Phi (\omega \cos \theta +\phi \sin \theta)\sin^{\alpha -2} \theta d\theta,\quad \phi \in S_{\omega}^{m-2}$ and $$S_{\omega}^{m-2}$$ is a unit sphere in the plane orthogonal to $$\omega \in S^{m-1}$$. The similar asymptotics is obtained for the forward scattering amplitude. The proofs of these results is reduced to the so-called eikonal approximation for the phase $$\phi$$ (x)$$of$$ the solution $$\psi$$ of the Schrödinger equation $$-\Delta \psi =k^ 2(1-\epsilon q)\psi$$, $$\epsilon =gk^{-2}$$, as $$k\to \infty$$. It suffices to take the first term of the formal series for $$\phi$$ (x) to obtain above asymptotics. The main body of the paper is devoted to the justification of validity of this procedure for the potentials of the considered class.
Reviewer: L.Pastur

##### MSC:
 35P25 Scattering theory for PDEs 81U05 $$2$$-body potential quantum scattering theory 34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations 47A40 Scattering theory of linear operators
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