| 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:
$$
\begin{equation}
-h^2 \Delta \psi+V(x) \psi=E\psi, \quad V=\gamma|x|^{-1}, \quad \psi \to e^{ikx_1}, \quad \ \ x_1\to -\infty,\quad x\in \mathbb{R}^3_x;
\end{equation}
\tag{1}
$$
here $h$, $\gamma$, and $k$ are positive parameters and $E=k^2$. The function $\psi$ must satisfy the Sommerfeld radiation conditions as $|x|\to \infty$; 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 $\psi$ can explicitly and globally be expressed in terms of the Airy function $\operatorname{Ai}$ and its derivative $\operatorname{Ai}'$ 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 $K^h_{\Lambda^n}\cdot 1$ (applied to $1$) on the $n$-dimensional (invariant) Lagrangian manifold $\Lambda^n$ formed by trajectories of the Hamiltonian system with Hamiltonian $H=p^2+V(x)$ in a $2n$-dimensional phase space that come out of a suitable $(n-1)$-dimensional plane $\widetilde\Lambda^{n-1}$. 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 $\widetilde\Lambda^{2}$ 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 $\Gamma$ onto the physical space are hyperbolae lying in planes passing through the origin. These hyperbolae have asymptotes, which are their limits as $t\to\pm\infty$. We select Keplerian orbits so that the corresponding asymptotes are orthogonal to the plane ($x_1=0$, $x_2=\alpha_2=\eta\cos\theta$, $x_3=\alpha_3=\eta\sin\theta$), and the corresponding momentum vector tends to the vector with components $(k,0,0)$ as $ t\to-\infty$. Then we obtain a family of orbits $P$, $X$ depending on $t$ (or some analogue of time) and the parameters $(\eta,\theta)$. 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 $\sigma$. This produces the Lagrangian manifold $\Lambda^3=\{p=P(\sigma,\eta,\theta), x= X(\sigma,\eta,\theta),\sigma \in (0,\infty),\eta\in (0,\infty), \theta \in S^1\}$, where the vectors $X$ and $P$ have components defined by the equalities (where $\mathbf{n}_2(\theta)=\cos\theta$, and $\mathbf{n}_3(\theta)= \sin\theta$)
$$
\begin{equation*}
X_1=\frac{\gamma}{2k^2} \biggl( \frac{\sigma \eta^2}{2}- \frac{(\sigma+1)^2}{2\sigma} \biggr),\qquad X_{2,3}=\frac{\gamma}{2k^2}\,\eta (\sigma+1) \,\mathbf{n}_{2,3}(\theta)
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
P_1=k\,\frac{-\sigma+\sigma \eta^2+1/\sigma} {\sigma+\sigma \eta^2+1/\sigma+2}\,, \qquad P_{2,3}=k\,\frac{2 \eta \sigma}{\sigma+\sigma \eta^2+1/\sigma+2}\, \mathbf{n}_{2,3}(\theta).
\end{equation*}
\notag
$$
Here $t=\dfrac{\gamma}{4k^3}\biggl(\dfrac{\sigma+\sigma\eta^2-1/\sigma}{2}+ \log \sigma+\dfrac{\log(1+\eta^2)}{2}\biggr)$. As $t\to -\infty$, we have
$$
\begin{equation*}
\begin{gathered} \, \sigma \to 0,\quad X_1\to -\infty, \quad X_{2,3} \to \frac{\gamma \eta}{2k^2}\mathbf{n}_{2,3}(\theta)= X_{2,3}^{\lim},\quad P_1\to k,\quad P_{2,3}\to 0. \end{gathered}
\end{equation*}
\notag
$$
Now we describe the objects involved in the asymptotic representation of the solution of (1) presented in the form $\psi_{\rm as}=K^h_{\Lambda^3}\cdot 1$.
The invariant measure $\mu$ on $\Lambda^3$ is defined by
$$
\begin{equation*}
\mu=d X_2^{\lim} \wedge dX_3^{\lim} \wedge dt=\biggl[\frac{\gamma^3}{32 k^7}\biggr] \eta\biggl(\eta^2+\biggl(1+\frac1{\sigma}\biggr)^2\biggr)\,d\eta\wedge d \theta \wedge d\sigma,
\end{equation*}
\notag
$$
and the Jacobian of the projection of $\Lambda^3$ onto $\mathbb R^3_x$ is $J=\mu^{-1}\,d X_1 \wedge d X_2 \wedge d X_3=2k(1-\sigma^2)$. We fix the (non-singular) central point with coordinates $\sigma=1-0$, $\eta=0$, and $\theta=0$ on $\Lambda^3$. The Lagrangian singularity on $\Lambda^3$, defined by $\sigma=1$, is a simple caustic (a fold), which has the form of the paraboloid of revolution $x_1=[k^2/(4\gamma)](x_2^2+x_3^2)-\gamma/k^2$. The projection $\pi_x\Lambda^3$ of $\Lambda^3$ onto $\mathbb{R}^3_x$ 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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