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UDC 517.37
MATHEMATICS
L. A. SAKHNOVICH
THE INTEGRAL OVER A PARABOLOID AND THE FIRST BORN APPROXIMATION
(Presented by Academician P. S. Novikov on 14 VIII 1965)
Consider the equation
\[ \Delta\psi+k^2\psi=qe^{ikx}. \tag{1} \]
As is known, a particular solution of this equation has the form
\[ \psi_1(x,y,z)=-\frac{1}{4\pi}\int e^{ik(x'-r')}q(x',y',z')\frac{dv'}{r'}, \tag{2} \]
where the integral is taken over all space, \(dv'=dx'dy'dz'\), \(r'^2=(x-x')^2+(y-y')^2+(z-z')^2\). In quantum mechanics, \(\psi_1\) gives the correction of the first Born approximation to the wave function (1). In the present paper we study in detail the dependence of \(\psi_1\) on \(q\) and \(k\) as \(k\to\infty\).
Equality (2) can be written in the form
\[ \psi_1(x,y,z)=-\frac{1}{4\pi}e^{ikx}\int e^{ik(s+r)}q(x+s,y+t,z+u)\frac{dv}{r}, \tag{3} \]
where \(r^2=s^2+t^2+u^2\), \(dv=ds\,dt\,du\).
In addition to the Cartesian coordinates of the point \(Q(s,t,u)\), introduce two more new coordinate systems \(Q(\rho,\varphi,\theta)\) and \(Q(\rho,s,\psi)\). Here \(\varphi\) and \(\theta\) are defined as in the spherical system, \(\rho=s+r\), \(\operatorname{tg}\psi=u/t\). Relation (3) can be represented as follows:
\[ \psi_1(x,y,z)=-\frac{1}{2}e^{ikx}\int_0^\infty e^{ik\rho} I(q,P,\rho)\,d\rho. \tag{4} \]
Here, to define \(I(q,P,\rho)\) at the point \(P(x,y,z)\), we use either of two formulas:
\[ I(q,P,\rho)=\frac{1}{2\pi}\int_0^{2\pi}\int_0^\pi \frac{\rho\sin\theta}{(1+\cos\varphi\sin\theta)^2} q\left(x+\frac{\rho\sin\theta\cos\varphi}{1+\cos\varphi\sin\theta}, y+\frac{\rho\sin\theta\sin\varphi}{1+\cos\varphi\sin\theta}, z+\frac{\rho\cos\theta}{1+\cos\varphi\sin\theta}\right) d\theta\,d\varphi, \tag{5} \]
\[ I(q,P,\rho)=\frac{1}{2\pi}\int_{-\infty}^{\rho/2}\int_0^{2\pi} q\left(x+s,\;y+\sqrt{\rho^2-2\rho s}\cos\psi,\;z+\sqrt{\rho^2-2\rho s}\sin\psi\right) d\psi\,ds. \tag{6} \]
The expression \(I(q,P,\rho)\) is the integral of the function \(q\) over the paraboloid of revolution \(s+r=\rho\) with focus at the point \(P\).
1. Basic properties of the integral over a paraboloid.
Introduce the notation \(I(q,\rho)=I(q,P,\rho)\) if \(x=y=z=0\).
- If the functions \(q(s,t,u)\) and \(f=s\,\partial q/\partial s+t\,\partial q/\partial t+u\,\partial q/\partial u+q\) are integrable over the paraboloid, then
\[ \frac{d}{d\rho}I(q,\rho)=\frac{1}{\rho}I(f,\rho). \]
The stated property is obtained as a result of differentiating both sides of (5).
- Let \(q(s,t,u)=a(s,r)t^m u^n\), where \(a(s,\rho-s)s^k\in L(-\infty,\rho/2)\) \((0\le k\le (m+n)/2)\), and \(m\) and \(n\) are nonnegative integers. Then
the equality holds
\[ I(q,\rho)=0, \tag{7} \]
if at least one of the numbers \(m\) or \(n\) is odd. If, however, \(m\) and \(n\) are even numbers, then
\[ I(q,\rho)= \frac{(n-1)!!(m-1)!!}{2^{(m+n)/2}((m+n)/2)!} \int_{-\infty}^{\rho/2} a(s,\rho-s)(\rho^2-2\rho s)^{(m+n)/2}\,ds . \tag{8} \]
Putting \(n=0,\ m=0\) in (8), we obtain
\[ I(q,\rho)=\int_{-\infty}^{\rho/2} a(s,\rho-s)\,ds,\qquad \text{where } q(s,t,u)=q(s,r). \tag{9} \]
- Suppose that the relation
\[ q(s,t,u)= \sum_{\substack{k,l\ge0\\ k+l\le n}} a_{k,l}(s)t^k u^l \frac{1}{k!l!} +q_n(s,t,u), \]
holds, where
\[ a_{kl}(s)= \left. \frac{\partial^{k+l}}{\partial t^k\partial u^l}q \right|_{t=0,\ u=0}, \qquad a_{k,l}(s)s^m\in L(-\infty,\rho/2),\quad 0\le m\le (k+l)/2, \]
and
\[ \lim_{n\to\infty} I(q_n,\rho)=0. \]
Then the equality
\[ I(q,\rho)= \sum_{k,l=0}^{\infty} \frac{1}{2^{2(k+l)}(k+l)!k!l!} \int_{-\infty}^{\rho/2} a_{2k,2l}(s)(\rho^2-2\rho s)^{k+l}\,ds \tag{10} \]
is valid.
- If the function \(q(s,t,u)\) is continuous and satisfies the inequality
\[ \left|q\left(s,\sqrt{\rho^2-2\rho s}\cos\psi, \sqrt{\rho^2-2\rho s}\sin\psi\right)\right| \le \varphi(s), \qquad s\le0,\quad \varphi(s)\in L(-\infty,0), \]
then
\[ \lim_{\rho\to0} I(q,\rho)= \int_{-\infty}^{0} q(s,0,0)\,ds . \]
- Suppose that the function \(q(s,t,u)\) is bounded and that the relation
\[ |q(s,t,u)-\varphi_1(s)| \le \varphi_2(s)(t^2+u^2)^\alpha,\qquad s\le0, \]
holds, where
\[ \varphi_1(s)\in L(-\infty,0),\qquad \varphi_2(s)s^\beta\in L(-\infty,0),\quad 0\le\beta\le\alpha,\quad \alpha\le \tfrac12 . \]
Then
\[ I(q,\rho)=\int_{-\infty}^{0}\varphi_1(s)\,ds+O(\rho^\alpha) \quad\text{as }\rho\to0. \]
- Suppose that \(q(s,t,u)\) is continuous at the point \(O(0,0,0)\) and the equality
\[ q(s,t,u)= \sum_{\substack{k,l\ge0\\ k+l\le2}} a_{kl}(s)t^k u^l \frac{1}{k!l!} +q_2(s,t,u), \]
holds, where
\[ a_{k,l}(s)= \left. \frac{\partial^{k+l}}{\partial t^k\partial u^l}q \right|_{t=0,\ u=0}, \qquad a_{kl}(s)s^m\in L(-\infty,\rho/2),\quad 0\le m\le \]
\[ \le (k+l)/2,\qquad |q_2(s,t,u)|\le \varphi(s)(t^2+u^2)^\alpha,\qquad s^m\varphi(s)\in L(-\infty,\rho/2), \]
\[ 0\le m\le\alpha,\qquad \alpha>1. \]
Then
\[ I(q,\rho)= \int_{-\infty}^{0} q(s,0,0)\,ds +\frac{\rho}{2} \left[ q(0,0,0) - \int_{-\infty}^{0} \left. \left(\frac{\partial^2}{\partial t^2} +\frac{\partial^2}{\partial u^2}\right)q \right|_{t=0,\ u=0} s\,ds +o(1) \right]. \tag{11} \]
- From the inequality \(|q(s,t,u)|\le \varphi(r)\) it follows that
\[ |I(q,\rho)|\le \int_{\rho/2}^{\infty}\varphi(r)\,dr . \tag{12} \]
§ 2. Integral over a paraboloid of a function equal to zero outside a convex body
According to Property 1, for smooth functions \(q(s,t,u)\) there exists the derivative \(\frac{d}{d\rho} I(q,\rho)\) \((\rho>0)\). We shall now clarify the conditions for existence and the method of computing \(\frac{d}{d\rho} I(q,\rho)\) in the case when \(q(s,t,u)\) has a discontinuity on some convex surface.
Let \(\Phi(s,t,u)=0\) be the equation of a convex surface \(S\), dividing all space into two nonintersecting parts \(R_1\) and \(R_2\) (\(R_1\) is the convex part). In what follows we shall assume that \(\Phi(s,t,u)\) has continuous first partial derivatives.
Denote by \(\chi(s,t,u)\) the characteristic function of the closure of the set \(R_1\), and by \(\gamma_\rho\) the line of intersection of the surface \(S\) with the paraboloid \(\rho=s+r\). Define on \(\gamma_\rho\) the function
\[ \Delta(s,t,u)=\frac{\partial \Phi}{\partial s}\left(\sqrt{s^2+t^2+u^2}-s\right)-t\frac{\partial \Phi}{\partial t}-u\frac{\partial \Phi}{\partial u}. \tag{13} \]
If at some point \(M(s,t,u)\) of the line \(\gamma_\rho\) the equality \(\Delta(s,t,u)=0\) holds, this means that the tangent to the meridian of the paraboloid at the point \(M\) belongs to the plane tangent to the surface \(S\) at the point \(M\).
Theorem 1. Let on \(\gamma_\rho\) the inequality
\[ |\Delta(s,t,u)|\geq \delta>0 \tag{14} \]
hold.
If the functions \(q\chi\), \(f\chi=(s\,\partial q/\partial s+t\,\partial q/\partial t+u\,\partial q/\partial u+q)\chi\) are integrable over the paraboloid and \(q\) is bounded on \(\gamma_\rho\), then the relation
\[ \frac{d}{d\rho} I(q\chi,\rho)= \]
\[ =\frac{1}{\rho}I(f\chi,\rho)+\frac{1}{2\pi\rho}\int_{\gamma_\rho} \left(s\frac{\partial \Phi}{\partial s}+t\frac{\partial \Phi}{\partial t}+u\frac{\partial \Phi}{\partial u}\right) q(s,t,u)\varepsilon(s,t,u)\frac{\rho-2s}{\Delta(s,t,u)}\,d\psi, \tag{15} \]
where \(\varepsilon(P)=-1\), if \(P(s,t,u)\) is a point of entry into the body \(R_1\) of the meridian of the paraboloid \(\rho=str\), and \(\varepsilon(P)=1\), if \(P\) is a point of exit.
§ 3. Asymptotics of \(\psi_1\) as \(k\to\infty\)
Theorem 2. Let the inequalities
\[ |q(x+s,y+t,z+u)|\leq \varphi_1(r,P), \qquad |f(x,y,z;s,t,u)|\leq \varphi_2(r,P) \]
hold,
\[ (r^2=s^2+t^2+u^2), \]
where
\[ f(x,y,z;s,t,u)=q(x+s,y+t,z+u)+ \]
\[ +(s\,\partial/\partial s+t\,\partial/\partial t+u\,\partial/\partial u)q(x+s,y+t,z+u), \]
\[ \int_0^\infty r^n\varphi_k(r,P)\,dr<\infty \quad (k=1,\ n=0,1;\ k=2,\ n=0). \]
Then the formula holds
\[ \psi_1(x,y,z)=\frac{1}{2ik}e^{ikx} \left[ \int_{-\infty}^{x} q(s,y,z)\,ds+ \int_0^\infty e^{ik\rho}\frac{1}{\rho}I(f,P,\rho)\,d\rho \right]. \]
Assume in addition that the following conditions are satisfied.
- The integral exists
\[ \int_0^\infty \varphi_2(r,P)|\ln r|\,dr<\infty. \]
- For some \(a\) (\(a>0\)) the inequality
\[ |f(x,y,z;s,t,u)-f(x,y,z,s,0,0)|\leq a(s)(t^2+u^2)^\alpha,\qquad s\leq 0, \]
holds, where \(a(s)s^\beta\in L(-\infty,0)\), \(0\leq \beta\leq \alpha\).
Then the integral
\[
\int_0^\infty e^{ik\rho}\frac{1}{\rho}I(f,P,\rho)\,d\rho
\]
converges absolutely and, consequently, tends to zero as \(k\to\infty\).
Thus (see (1)),
\[
\psi_1(x,y,z)=\frac{1}{2ik}e^{ikx}\left[\int_{-\infty}^{x}q(s,y,z)\,ds+o(1)\right],\qquad k\to\infty.
\]
Consider a bounded closed convex surface \(S\), whose equation is \(\Phi(x,y,z)=0\), where \(\Phi(x,y,z)\) has continuous partial derivatives of first order. We shall say that \(P(x,y,z)\) is a point of focus type of the surface \(S\) if the common points \(Q(x',y',z')\) of the surface \(S\) and of some paraboloid
\[
(y'-y)^2+(z'-z)^2=\rho^2-2\rho(x'-x)
\]
form a set of positive planar measure. Let the function \(q_1(x,y,z)\) have continuous partial derivatives of first order and, for some \(K\) and \(\alpha\), satisfy the inequality
\[
|s|\left|\frac{\partial}{\partial s}q_1(x+s,y+t,z+u)-\frac{\partial}{\partial s}q_1(x+s,y,z)\right|
\leq K(t^2+u^2)^\alpha,\quad s\leq 0,\ \alpha>0.
\]
Finally, put \(q(x,y,z)=q_1(x,y,z)\chi(x,y,z)\), where \(\chi(x,y,z)\) is the characteristic function of \(R_1\) (see item 2).
If \(P(x,y,z)\in R_1\) and \(P\) is not a point of focus type of the surface \(S\), then as \(k\to\infty\)
\[
\psi_1(x,y,z)=\frac{1}{2ik}e^{ikx}\left[\int_{x_0}^{x}q(s,y,z)\,ds+o(1)\right],
\]
where \(\Phi(x_0,y,z)=0\) and \(x_0<x\).
Odessa Electrotechnical
Institute of Communications
Received
11 VIII 1965
REFERENCES
- L. D. Landau, E. M. Lifshitz, Quantum Mechanics, Moscow, 1963.