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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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