MATHEMATICAL PHYSICS

V. MASLOV

THE SCATTERING PROBLEM IN THE QUASICLASSICAL APPROXIMATION

(Presented by Academician N. N. Bogolyubov on 28 I 1963)

Let us consider a solution \(\psi(x)\) of the Schrödinger equation (for simplicity we take \(m=1/2\))

\[ -h^2 \Delta \psi + v(x)\psi = E\psi,\qquad x=x_1,\ldots,x_n, \tag{1} \]

where the potential \(v(x)\) is a holomorphic function of all the real arguments \(x_1,\ldots,x_n\), and \(v(x)=O(|x|^{-2n})\) as \(|x|\to\infty\). Suppose \(\psi(x)\) satisfies the condition

\[ \psi=\exp\left[\frac{i}{h}\sqrt{E}x_1\right] +|x|^{(1-n)/2} f(\varphi)\exp\left[\frac{i}{h}\sqrt{E}\,|x|\right] +o\left(|x|^{(1-n)/2}\right), \tag{2} \]

where \(\varphi=\varphi_1,\varphi_2,\ldots,\varphi_{n-1}\) are the angular coordinates in the spherical coordinate system in the \(n\)-dimensional case.

The problem is to find the asymptotics of the functions \(\psi(x)\) and \(f(\varphi)\) as \(h\to0\). The asymptotics of the function \(f(\varphi)\) were sought in papers \((^{1-3})\). However, as was shown in \((^{4,5})\) on the example of the one-dimensional case, the formulas in \((^{1-3})\) are incorrect. The formulas given below agree in the one-dimensional case with the results of \((^{4,5})\). Their derivation is based on the method set forth in \((^{6})\).

First of all, let us formulate the problem in classical mechanics which will correspond to the given quantum problem.

1. Consider the complex vector \(z(t): z=z_1,\ldots,z_n\) of the complex argument \(t\), which serves as a solution of the system of equations of classical mechanics (where we put \(m=1/2\)): \(\frac12 \dot z_i=-\partial v(z)/\partial z_i\), \(i=1,\ldots,n\), and satisfies, as \(\operatorname{Re} t\to-\infty\), the conditions \(z_i(t)\to z_i^0\) for \(i>1\) and \(z_1(t)\to 2\sqrt{E}\,t\), where \(z_i^0\) are arbitrary complex numbers. In addition, it is required that there exist a complex point \(t=t'\) at which the trajectory \(z(t)\) would pass through the given fixed point \(x\) on the real hyperplane: \(z(t')=x\). The numbers \(z_i^0,\ i=2,\ldots,n\), will, obviously, depend on \(x\).

These conditions determine, generally speaking, more than one trajectory.

1. Example 1. Consider the case \(n=1\) and the potential shown in Fig. 1. It has one well. Let the point \(x'\) lie to the left of the point \(a\). Then, in particular, the paths shown in Fig. 1 will pass through this point. Path \(I\) is a straight line. Path \(II\) has undergone reflection from the “turning point” \(a\), path \(III\) has undergone reflection from the “turning point” \(b\). The complex points \(t_i'\) corresponding to them are equal to

\[ t_1'(x')=\frac12\int_{-\infty}^{x'}(p^{-1}-E^{-1/2})\,dx+\frac12 x'E^{-1/2},\qquad t_2'=t_1'(a)-\frac12\int_a^{x'}p^{-1}\,dx, \]

\[ t_3'=t_2'+i\int_a^b |p|^{-1}\,dx, \]

where

\[ p_m=\sqrt{E-v(x)}. \]

1. We denote by \(J\) the Jacobian of the passage from the current coordinates of the trajectory \(z_1(t,z_2^0,\ldots,z_n^0),\ldots,z_n(t,z_2^0,\ldots,z_n^0)\) to the coordinates \(t,z_2^0,\ldots,z_n^0\):
   \[ J=D(z_1,\ldots,z_n)/D(t,z_2^0,\ldots,z_n^0). \]
   We note that for \(\operatorname{Re}t=-\infty\) the Jacobian is equal to \(2\sqrt{E}\). In the one-dimensional case
   \[ J=\dot z=2\sqrt{E-v(x)}. \]

2. A point \(z(t_1)\), \(t_1\), is called a focus on the trajectory \(z(t)\) if the Jacobian \(J\) vanishes at this point. (Thus, in Example 1 the points \(a\) and \(b\) are foci.) In what follows we shall assume that the point \(x\) is not a focus for any of the trajectories arriving at it.

Fig. 1

1. The phase \(\gamma(t_1)\) of the path \(z(t)\) at the point \(t=t_1\) will mean \(\arg J\) at the point \(t=t_1\), under the condition that \(\arg J=0\) for \(\operatorname{Re}t=-\infty\). The phase of a path depends on the entire path \(z(t)\). The total phase of the path \(\gamma(x)\) will mean the phase of the path at the point \(t'\) at which \(z(t')=x\). (In Example 1 the total phases of the three paths are respectively \(0,\pi,3\pi\), if the foci are bypassed clockwise.)

2. The action \(S(x)\) at the point \(x\) along the path \(z(t)\) is equal to
   \[ S(x)=\frac12 \dot z(-\infty)x+\frac12 \int_{z(-\infty)}^{x}[\dot z(t)-\dot z(-\infty)]\,dz(t), \]
   where \(z(-\infty)\) and \(\dot z(-\infty)\) denote the limits of \(z(t)\) and \(\dot z(t)\) as \(\operatorname{Re}t\to-\infty\).

3. Two paths \(z(t)\) and \(\tilde z(t)\) passing through the point \(x\) are called homotopic to one another if their actions and total phases are identically equal in some neighborhood of the points \(x,t\) and \(z_2^0,\ldots,z_n^0\).

4. A path \(z(t)\) is called determined if there exists a path \(\tilde z(t)\), homotopic to \(z(t)\), for which the real and imaginary parts of the action do not decrease along the path from \(\operatorname{Re}t=-\infty\) to \(x\).

5. Consider the set\(^*\) of determined non-homotopic paths passing through the point \(x\), and assign to each of them, and accordingly to their actions, phases, and Jacobians, two indices so that the inequalities
   \[ \operatorname{Im}S_{ik}<\operatorname{Im}S_{jm}\quad \text{for } i<j\ (i,j=0,1,\ldots) \]
   and, for all \(k,m\),
   \[ \operatorname{Re}S_{ik}<\operatorname{Re}S_{im}\quad \text{for } k>m\ (k,m=0,1,\ldots) \]
   are satisfied.

6. We shall call the point \(x\) resonant if, for some \(k\), the sum of the series
   \[ \sigma_k=\sum_{j=0}^{\infty}2E^{1/2}J_{kj}^{-1/2}\exp\left[\frac{i}{h}S_{kj}-\frac{i}{2}\gamma_{kj}(x)\right] \]
   is infinite in modulus.

7. If the point \(x\) is not a focus for any of the trajectories passing through it, and is not a resonant point, then the solution \(\psi(x)\) of equation (1), satisfying conditions (2), can be represented in the form
   \[ \psi(x)=\sigma_0(1+O(h)), \]
   where the summation of the series \(\sigma_0\) is performed in the Abel sense.\(^ {**}\)

8. Example 2. Let \(n=1\). Consider the case of a potential with one well (Fig. 1). Let the point \(x\) lie to the right of the point \(e\). Let
   \[ p(x)=\sqrt{E-v(x)}. \]
   Then
   \[ S_{00}=i\int_a^b |p|\,dx+i\int_v^e |p|\,dx+\int_e^x p\,dx+\sqrt{E}x+ \]

\[ \phantom{.} \]

* It is countable; otherwise the point \(x\) would be a focus.

** This asymptotics will be uniform if the point \(x\) lies outside a certain \(h\)-independent neighborhood of the foci and resonant points, and \(E\) lies outside a neighborhood of the quasistationary levels.

\[ +\int_{-\infty}^{a}\lvert p-\sqrt{E}\rvert\,dx+\int_{\delta}^{\varepsilon}p\,dx;\qquad J_{jk}=2p(z_{jk}). \]
Put
\[ \alpha_{jk}=\exp[i\gamma_{jk}]=J_{jk}/|J_{jk}|;\qquad \beta=\int_{\delta}^{\varepsilon}p\,dx. \]
When the path \(z_{00}(t)\) passes through the foci \(a,b,\varepsilon,g\), the coefficient \(\alpha_{00}\) changes, respectively, as
\[ \alpha_{00}=1\to i\to 1\to i\to 1. \]
Hence
\[ \gamma_{00}=0\to \pi/2\to 0\to \pi/2\to 0. \]

The path \(z(t)\) passes twice through the foci \(b\) and \(\varepsilon\): after being reflected from the point \(\varepsilon\), and then from the point \(b\). Therefore
\[ \alpha_{01}=1\to i\to 1\to -1\to 1\to i\to 1. \]
Consequently,
\[ \gamma_{01}(x)=0\to \pi/2\to 0\to \pi\to 2\pi\to {}^{5}/_{2}\,\pi\to 2\pi. \]
The action is \(S_{01}=S_{00}+2\beta\). Similarly,
\[ S_{0k}=S_{00}+2k\beta;\qquad \gamma_{0k}(x)=2k\pi. \]
By virtue of 11 we have
\[ \Psi(x)\simeq E^{1/2}p^{-1/2}\exp\left[\frac{i}{h}S_{00}\right] \sum_{k=0}^{\infty}\exp\left[ik\left(\frac{2\beta}{h}-\pi\right)\right] = E^{1/2}p^{-1/2}\exp\left(\frac{i}{h}S_{00}\right)\times \]
\[ \times\left\{1-\exp\left(\frac{2i\beta}{h}-i\pi\right)\right\}^{-1}. \]
If \(2\beta/h-\pi=2\pi n\), then \(x\) is a resonance point. Thus resonance occurs if the energy \(E\) satisfies the equation
\[ \beta=h\pi(n+{}^{1}/_{2}), \]
i.e., we are on a quantization level.

1. Let us now write the asymptotics of the function \(f(\varphi)\). Let the point \(x\) tend to \(\infty\) in the direction determined by the angular coordinates \(\varphi=\varphi_1,\ldots,\varphi_{n-1}\). Denote
   \[ 2\widetilde S = z(t_0)(\dot z(-\infty)-\dot z(+\infty)) + \int_{z(-\infty)}^{z(t_0)}[\dot z(t)-\dot z(-\infty)]\,dz(t) + \]
   \[ + \int_{z(t_0)}^{z(+\infty)}\{\dot z(t)-\dot z(+\infty)\}\,dz(t); \]
   where \(t_0\) is an arbitrary point;
   \[ 4\Delta=E^{-1}\lim_{\operatorname{Re}t\to\infty}|z|^{1-n}J,\qquad \widetilde\gamma=\lim_{\operatorname{Re}t\to\infty}\gamma. \]

Assume that \(|\Delta|>0\) for all paths of this type. Consider the set of all deterministic paths, not homotopic to one another, passing as \(\operatorname{Re}t\to\infty\) in the direction of the angle \(\varphi\). As before, provide the paths with two indices and define, analogously to the preceding, the resonance angle.

Let the angle \(\varphi\) be nonresonant; then
\[ f(\varphi)=(1+O(h))\sum_{k=0}^{\infty} \Delta_{0k}^{-1/2}\exp\left(\frac{i}{h}\widetilde S_{0k}-\frac{i}{2}\widetilde\gamma_{0k}\right), \tag{3} \]
where the summation is carried out in the Abel sense.

1. Example 3. Consider above-barrier reflection in the one-dimensional case: \(E>v(x)\). Let \(a_i\) be the complex roots of the equation \(E-v(z)=0\). The solutions of Newton’s equation satisfying the conditions \(z\to 2\sqrt E\,t\) as \(\operatorname{Re}t\to-\infty\) and \(z\to-\infty\) as \(\operatorname{Re}t\to+\infty\) obviously go around at least one of the points \(a_i\). Suppose that the circuit around the point \(a_0\) gives the minimum of the imaginary part of \(\widetilde S\), and let \(a_0\) be a simple root. Then the circuit around \(a_0\) gives a change of phase of the Jacobian \(J=2\sqrt{E-v(z)}\) by \(\pi\). Therefore
   \[ f=\exp\left(\frac{i}{h}\widetilde S-\frac{i\pi}{2}\right) \]
   in accordance with \(({}^{4},{}^{5})\).

2. Example 4. Let \(n=3\), and let the potential depend only on
   \[ r=|x|:\quad v(x)=v(r). \]
   Pass to the spherical coordinate system \(r,\varphi,\theta\) of the current point \(z(t)\) and to the polar coordinate system \(\rho,\theta_0\) for the determination of the direction ...

initial point \(z_2^0, z_3^0\). Then
\(J=r^2\sin\theta\,\rho^{-1}D(r,\varphi)/D(\rho,t)\). It is not hard to verify\({}^{(7)}\) that
\(D(r,\varphi)/D(\rho,t)=\dot r\,\partial\varphi/\partial\rho\), where
\(\dot r=2\sqrt{E-v(r)-\rho^2Er^{-2}}\).

Let \(\{\rho(\varphi_0)\}\) be the set of complex solutions of the implicit equation

\[ \varphi_0=2\int_{r_0(\rho)}^{\infty} \rho E^{1/2}r^{-2}\dot r^{-1}(\rho,r)\,dr, \]

where \(r_0(\rho)\) is a simple root of the equation
\(\dot r^{\,2}(\rho,r)=0\). From the set \(\{\rho(\varphi_0)\}\) choose the (let it be unique) solution \(\rho_0\) that gives a minimum of \(\operatorname{Im}\widetilde S\). The change in the phase of the Jacobian \(J\) in going around the point \(r_0(\rho_0)\) is equal to \(\pi\); therefore, by virtue of (3), one may write

\[ f(\varphi_0)\simeq -i\rho_0^{1/2}\sin^{-1/2}\theta\, (\partial\varphi_0/\partial\rho_0)^{-1/2} \exp\left[ \frac{2i}{h}\left( \frac{1}{2}\int_{r_0}^{\infty}\dot r(\rho_0,r)\,dr +r_0E^{1/2} \right) \right]. \]

In conclusion we note that all the results of the article carry over directly to the case of a complex potential \(v(x)\).

Moscow State University
named after M. V. Lomonosov

Received
2 I 1963

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