| Russian Mathematical Surveys |
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Brief communications Keplerian orbits and global asymptotic solution in the form of an Airy function for the scattering problem on a repulsive Coulomb potential S. Yu. Dobrokhotova, S. B. Levinb, A. A. Tolchennikova a Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow b Saint Petersburg State University
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     We consider the following scattering problem: here h, γ, and k are positive parameters and E=k2. The function ψ must satisfy the Sommerfeld radiation conditions as |x|→∞; we do not need this condition. The exact solution of this problem is well known ([1], formulae (5.1) and (5.2)); it is expressed in terms of a confluent hypergeometric function. The aim of this note is to show that for small h the (quasiclassical) asymptotics of the solution ψ can explicitly and globally be expressed in terms of the Airy function Ai and its derivative Ai′ of a certain complex argument defined given by the well-known Keplerian orbits (for instance, see [2]).The quasiclassical asymptotics of the scattering problem for the n-dimensional Schrödinger equation and a (smooth) potential V(x) with compact support was studied in [3] and [4]. It can be represented in the form of the Maslov canonical operator KhΛn⋅1 (applied to 1) on the n-dimensional (invariant) Lagrangian manifold Λn formed by trajectories of the Hamiltonian system with Hamiltonian H=p2+V(x) in a 2n-dimensional phase space that come out of a suitable (n−1)-dimensional plane ˜Λn−1. In the case of problem (1), finding the asymptotic behaviour of the solution follows a similar pattern. In the case of a problem with Coulomb potential, there is a significant difference from the problem where the potential has a compact support: the plane ˜Λ2 in the phase space must be ‘taken to the appropriate infinity’. We implement this idea as follows. The corresponding Hamiltonian system is integrable, and its solutions are well-known Keplerian orbits, whose projections Γ onto the physical space are hyperbolae lying in planes passing through the origin. These hyperbolae have asymptotes, which are their limits as t→±∞. We select Keplerian orbits so that the corresponding asymptotes are orthogonal to the plane (x1=0, x2=α2=ηcosθ, x3=α3=ηsinθ), and the corresponding momentum vector tends to the vector with components (k,0,0) as t→−∞. Then we obtain a family of orbits P, X depending on t (or some analogue of time) and the parameters (η,θ). Since the dependence of Keplerian orbits on time t is described parametrically, in parametrizing the Lagrangian manifold it is more convenient to replace t by a more suitable parameter σ. This produces the Lagrangian manifold Λ3={p=P(σ,η,θ),x=X(σ,η,θ),σ∈(0,∞),η∈(0,∞),θ∈S1}, where the vectors X and P have components defined by the equalities (where n2(θ)=cosθ, and n3(θ)=sinθ) and Here t=γ4k3(σ+ση2−1/σ2+logσ+log(1+η2)2). As t→−∞, we have Now we describe the objects involved in the asymptotic representation of the solution of (1) presented in the form ψas=KhΛ3⋅1.The invariant measure μ on Λ3 is defined by and the Jacobian of the projection of Λ3 onto R3x is J=μ−1dX1∧dX2∧dX3=2k(1−σ2). We fix the (non-singular) central point with coordinates σ=1−0, η=0, and θ=0 on Λ3. The Lagrangian singularity on Λ3, defined by σ=1, is a simple caustic (a fold), which has the form of the paraboloid of revolution x1=[k2/(4γ)](x22+x23)−γ/k2. The projection πxΛ3 of Λ3 onto R3x is the set \{x\in \mathbb{R}^3\colon x_1 \leqslant [k^2/(4\gamma)](x_2^2+x_3^2)- \gamma/k^2\}. Each interior point x\in \pi_x\Lambda^3 has the two preimages with coordinates \sigma_\pm, \eta_\pm, and \theta on \Lambda^3, where
\begin{equation*}
\begin{gathered} \, \sigma_\pm=z \pm \sqrt{z^2-1}\,, \\ \eta_\pm=\frac{2k^2}{\gamma}\,\frac{\sqrt{x_2^2+x_3^2}}{z+1\pm\sqrt{z^2-1}} \end{gathered}
\end{equation*}
\notag
and
\begin{equation*}
z(x)=\frac{k^2}{\gamma}\bigl(-x_1+\sqrt{x_2^2+x_3^2}\,\bigr)-1.
\end{equation*}
\notag
Outside a neighbourhood of the caustic \Lambda^3 can be covered by two non-singular (non-compact) domains, with Maslov indices set to be m_-=0 and m_+=1. The action function on \Lambda^3 is S=[\gamma/(2k)](-\log \sigma+\sigma \eta^2/2-1/(2\sigma)+\sigma/2). At the points x\in \pi_x\Lambda^3 it generates the two phases S_\pm(x)=[\gamma/(2k)]\bigl[-\log (z\pm \sqrt{z^2-1}\,)+ z \pm \sqrt{z^2-1}+1\bigr]+kx_1, and the WKB-asymptotics at interior points of \pi_x\Lambda^3 is as follows: \psi_{\rm as}(x)=\sum_\pm A_\pm(x) e^{(i/h)S_\pm(x)}, A_\pm(x)=e^{-i\pi m_\pm/2}/[\sqrt{2k}\,(z^2-1)^{1/4}(\sqrt{z+1}\pm \sqrt{z-1}\,)] . Here the term with subscript + corresponds to the incoming wave and the one with subscript -, to the reflected wave.
Another important feature of the problem considered consists in the fact that, using arguments from [5] and [6], we can globally express the asymptotics of the solution in terms of the functions \operatorname{Ai} and \operatorname{Ai}'. Consider the functions \Theta(x)=(S_++S_-)/2=k\bigl(x_1+\sqrt{x_2^2+x_3^2}\,\bigr)/2 for x\in \mathbb{R}^3_x and \Psi=(S_+- S_-)/2=[\gamma/(2k)][\sqrt{z^2-1}- \log (z+\sqrt{z^2-1}\,)] for x\in \pi_x\Lambda^3. Note that \Psi \sim [\gamma/(12k)](2(z-1))^{3/2} as z\to 1+0. Let \Phi and A_\pm be the functions defined by \Phi(x)=(3\Psi(x)/2)^{2/3} and A_\pm(x) =(3\Psi(x)/2)^{\pm1/6} [(z+1)/(z-1)]^{\pm1/4} /\sqrt{2k} for z(x)>1 and by \Phi(x)=(\gamma/k)^{2/3}(z-1)/2 and A_\pm(x)=(\gamma/k)^{\pm1/6}/\sqrt{2k} for z(x)\leqslant 1. Theorem. The leading term of the asymptotic solution \psi_{\rm as}=K^h_{\Lambda^3}\cdot 1 of problem (1) is described by the formula 
Citation in format AMSBIB
\Bibitem{DobLevTol23} |
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