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UDC 517.37
MATHEMATICS
L. A. SAKHNOVICH
THE INTEGRAL OVER AN ELLIPSOID AND THE SECOND BORN APPROXIMATION
(Presented by Academician I. M. Vinogradov, 11 IX 1965)
Let us write the equation
\[ \Delta \psi + k^2 \psi - q\psi = 0, \tag{1} \]
where \(\psi - e^{ikx} \to 0\) as \(r \to \infty\). The correction \(\psi_1\) of the first Born approximation to the wave function \(\psi\) can, as is known, be written in the form
\[ \psi_1(x,y,z)=-\frac{1}{4\pi}\int e^{ik(x'+r')}q(x',y',z')\frac{dv'}{r'}, \]
\[ r'^2=(x-x')^2+(y-y')^2+(z-z')^2, \]
and the integral is taken over all space. In article (1) we studied the behavior of \(\psi_1\) as \(k\to\infty\), using the properties of the integral over a paraboloid.
Let us now write the second correction \(\psi_2\) to the wave function:
\[ \psi_2(P)=\frac{1}{4\pi}\int e^{ikx'}q(P')R(P,P')\,dv', \tag{2} \]
where
\[ R(P,P')=\frac{1}{4\pi}\int e^{ik(r'+r'')}q(P'')\frac{dv''}{r'r''}. \tag{3} \]
Here the points \(P,P',P''\) have, respectively, the coordinates \((x,y,z)\), \((x',y',z')\), \((x'',y'',z'')\); \(r'=PP'\), \(r''=P'P''\). The purpose of this article is to study the behavior of \(R(P,P')\) as \(k\to\infty\).
Relation (3) can be represented in the form
\[ R(P,P')=\int_c^\infty e^{2ik\rho} I(q,P,P',\rho)\,d\rho,\qquad c=\frac{PP'}{2}. \tag{4} \]
The integral \(I(q,P,P',\rho)\) is determined by either of two formulas:
\[ I(q,P,P',\rho)=\frac{1}{4\pi\rho} \int_{-\rho}^{\rho}\int_0^{2\pi} q\left( \frac{x+x'}{2}+a_{11}s+a_{12}t+a_{13}u,\, \frac{y+y'}{2}+a_{21}s+ \right. \]
\[ \left. {}+a_{22}t+a_{23}u,\, \frac{z+z'}{2}+a_{11}s+a_{32}t+a_{13}u \right)d\psi\,ds, \tag{5} \]
\[ I(q,P,P',\rho)=\frac{1}{4\pi} \int_0^{2\pi}\int_0^\pi q\left( \frac{x+x'}{2}+a_{31}s+a_{12}t+a_{33}u,\, \frac{y+y'}{2}+ \right. \]
\[ \left. {}+a_{21}s+a_{22}t+a_{23}u,\, \frac{z+z'}{2}+a_{31}s+a_{32}t+a_{33}u \right) D(\rho,\theta,\varphi)\,d\theta\,d\psi, \tag{6} \]
where the matrix \(A=\|a_{ij}\|\) is orthogonal and the vector \(\bar h[a_{11}a_{21}a_{31}]\) is parallel to the vector \(PP'\). In addition, in formula (5)
\[ t=\frac{\sqrt{(\rho^2-c^2)(\rho^2-s^2)}}{\rho}\cos\psi,\qquad u=\frac{\sqrt{(\rho^2-c^2)(\rho^2-s^2)}}{\rho}\sin\psi, \]
and in formula (6)
\[ s=r\cos\varphi\sin\theta,\qquad t=r\sin\varphi\sin\theta,\qquad u=r\cos\theta,\qquad r=\rho\sqrt{\frac{\rho^2-c^2}{\rho^2-c^2\cos^2\varphi\sin^2\theta}}, \]
\[ D(\rho,\theta,\varphi)= \frac{\rho^2\sqrt{\rho^2-c^2}\,\sin\theta} {\bigl(\rho^2-c^2\cos^2\varphi\sin^2\theta\bigr)^{1/2}}. \]
Let, in particular, \(q(P'')=q(r'r'')\). Then formula (5) gives
\[ I(q,\ P,\ P',\ \rho)=\frac{1}{2c}\int_{\rho^2-c^2}^{\rho^2} q(t)\frac{1}{\sqrt{\rho^2-t}}\,dt, \tag{7} \]
If \(q(P'')=q(r',r'')\), then
\[ I(q,\ P,\ P',\ \rho)=\frac{1}{2\rho}\int_{-\rho}^{\rho} q\left(\rho-\frac{c}{\rho}t,\ \rho+\frac{c}{\rho}t\right)\,dt. \tag{8} \]
The expression \(I(q,P,P',\rho)\) is the integral of the function \(q\) over an ellipsoid of revolution with foci at the points \(P\) and \(P'\).
1. Basic properties of the integral over an ellipsoid
Introduce the notation \(I(q,\rho,c)=I(q,P,P',\rho)\), \(\rho>c\), if \(y'=y=0\), \(z'=z=0\), \(x'+x=0\) \((c=|x|)\). The properties of the integral over an ellipsoid formulated below are in many respects similar to the properties of the integral over a paraboloid (1).
- If the functions \(q(s,t,u)\) and \(f=s\,\partial q/\partial s+t\,\partial q/\partial t+u\,\partial q/\partial u\) are bounded, then
\[ \frac{d}{d\rho}I(q,\rho,c) = \frac{1}{\rho(\rho^2-c^2)}I(fr'r'',\rho,c) + \frac{3\rho^2-c^2}{\rho(\rho^2-c^2)}I(q,\rho,c) - \frac{3}{\rho(\rho^2-c^2)}I(qr^2,\rho,c), \tag{9} \]
where
\[ r^2=s^2+t^2+u^2,\qquad r'=\rho-\frac{c}{\rho}s,\qquad r''=\rho+\frac{c}{\rho}s. \]
- Let \(q(s,t,u)=a(r',r'')t^n u^m\), where \(a(r',r'')\) is a bounded function and \(m\) and \(n\) are nonnegative integers. Then the equality
\[ I(q,\rho,c)=0, \tag{10} \]
holds if at least one of the numbers \(m\) or \(n\) is odd.
If, however, \(m\) and \(n\) are even numbers, then
\[ I(q,\rho,c)= \frac{(n-1)!!(m-1)!!}{2^{(m+n)/2}\left(\frac{m+n}{2}\right)!}\, \frac{1}{2\rho} \int_{-\rho}^{\rho} a\left(\rho-\frac{c}{\rho}s,\ \rho+\frac{c}{\rho}s\right) \times \]
\[ \times \left[ \frac{\sqrt{(\rho^2-c^2)(\rho^2-s^2)}}{\rho} \right]^{m+n} \,ds. \tag{11} \]
- Suppose the relation
\[ q(s,t,u)= \sum_{\substack{k,l\ge 0\\ k+l\le n}} a_{k,l}(s)t^k u^l\frac{1}{k!\,l!} + q_n(s,t,u) \]
holds, where the functions
\[ a_{k,l}(s)= \left. \frac{\partial^{k+l}}{\partial t^k\partial u^l}q \right|_{\substack{t=0\\ u=0}} \]
are bounded and \(\lim_{n\to\infty} I(q_n,\rho,c)=0\). Then the equality is true:
\[ I(q,\rho,c)= \sum_{k,l=0}^{\infty} \frac{1}{2^{2(k+l)}(k+l)!\,k!\,l!}\, \frac{1}{2\rho} \int_{-\rho}^{\rho} a_{2k,2l}(s) \left[ \frac{(\rho^2-c^2)(\rho^2-s^2)}{\rho^2} \right]^{k+l} \,ds. \]
- Let the function \(q(s,t,u)\) be bounded and have the form \(q(s,t,u)=a(r',r'')e^{\beta t+\gamma u}\). Then
\[ I(q,\rho,c)= \frac{1}{2\rho}\int_{-\rho}^{\rho} a\left(\rho-\frac{c}{\rho}s,\ \rho+\frac{c}{\rho}s\right) J_0 \left[ \frac{\sqrt{(\beta^2+\gamma^2)(\rho^2-c^2)(\rho^2-s^2)}}{\rho} \right]\,ds, \tag{12} \]
where \(J_0(z)\) is the Bessel function.
- If \(q(s,t,u)\) is a continuous function, then
\[ \lim_{\rho\to c} I(q,\rho,c)=\frac{1}{2c}\int_{-c}^{c} q(s,0,0)\,ds . \]
- Let the function \(q(s,t,u)\) be bounded and let the relation
\[ \left|q(s,t,u)-\varphi_1(s)\right|\leq \varphi_2(s)(t^2+u^2)^\alpha,\quad |s|\leq c,\quad 0<\alpha\leq \frac12, \]
hold, where the functions \(\varphi_1(s)\) and \(\varphi_2(s)\) are also bounded. Then
\[ I(q,\rho,c)=\frac{1}{2c}\int_{-c}^{c} q(s,0,0)\,ds +O\!\left[\left(\frac{\rho-c}{\rho}\right)^\alpha\right] \quad \text{as } \rho\to c . \tag{13} \]
By imposing additional smoothness requirements on \(q(s,t,u)\), one can find the subsequent terms of the expansion of \(I(q,\rho,c)\) in powers of \(\rho-c\) as \(\rho\to c\).
- Let the function \(q(s,t,u)\) be continuous and let the equality
\[ q(s,t,u)= \sum_{\substack{k,l\geq 0\\ k+l\leq 2}} a_{k,l}(s)\,t^k u^l\,\frac{1}{k!\,l!} +q_2(s,t,u), \]
hold, where the functions
\[
a_{k,l}(s)=
\left.
\frac{\partial^{k+l}}{\partial t^k\partial u^l}q
\right|_{\substack{t=0\\ u=0}}
\]
are bounded and \(|q_2(s,t,u)|\leq C(t^2+u^2)^\alpha\), \(\alpha>1\). Then
\[ I(q,\rho,c)= \frac{1}{2c}\int_{-c}^{c} q(s,0,0)\,ds +\frac{\rho-c}{c}\,\frac{q(c,0,0)+q(-c,0,0)}{2} - \]
\[ -\frac{\rho-c}{2c^2}\int_{-c}^{c}q(s,0,0)\,ds +\frac{\rho-c}{4c^2}\int_{-c}^{c} \left(\frac{\partial^2 q}{\partial t^2}+\frac{\partial^2 q}{\partial u^2}\right)_{\substack{t=0\\ u=0}} (c^2-s^2)\,ds +o(\rho-c)\quad \text{as }\rho\to c . \]
- Let \(|q(s,t,u)|\leq \varphi(r)\), where the function \(\varphi(r)\) decreases monotonically. Then
\[ |I(q,\rho,c)|\leq \varphi\!\left(\sqrt{\rho^2-c^2}\right). \]
The last inequality characterizes the behavior of \(I(q,\rho,c)\) as \(\rho\to\infty\).
2. Asymptotics of \(R(P,P')\) as \(k\to\infty\).
Theorem 1. Let the following conditions be satisfied:
-
The function \(q(x,y,z)\) has bounded first-order derivatives.
-
There exists a monotonically decreasing function \(\varphi(r)\) such that
\[ \left| q\!\left( \frac{x+x'}{2}+s,\, \frac{y+y'}{2}+t,\, \frac{z-z'}{2}+u \right) \right| \leq \varphi(r), \quad \varphi(r)\in L(-\infty,0), \]
\[ r^2=s^2+t^2+u^2 . \]
Then the equality
\[ R(P,P')= -\frac{e^{2ikc}}{4ikc}\int_l q(P'')\,dl -\frac{1}{2ik}\int_c^{\infty} e^{2ik\rho}\, \frac{d}{d\rho}I(q,P,P',\rho)\,d\rho, \]
\[ c=\frac{PP'}{2}, \tag{14} \]
holds, where the integration contour \(l\) coincides with the segment \(PP'\).
The expression \(\dfrac{d}{d\rho}I(q,P,P',\rho)\) in the right-hand side of (14) can be found by means of formula (9).
Suppose, additionally, that the following conditions are satisfied:
- There exists a monotonically decreasing function \(\psi(r)\) such that, for
\[ q_1(s,t,u)=q\!\left(\frac{x+x'}{2}+s,\frac{y+y'}{2}+t,\frac{z+z'}{2}+u\right), \]
the inequality
\[ |\partial q_1/\partial s|+|\partial q_1/\partial t|+|\partial q_1/\partial u| \leq \psi(r),\quad \psi(r)\in L(0,\infty). \]
- The function \(q_1(s,t,u)\) satisfies, for some \(K\) and \(\alpha > 0\), the Hölder condition. Then the integral
\[ \int_c^\infty e^{2ik\rho}\,\frac{d}{d\rho} I(q, P, P', \rho)\,d\rho \tag{15} \]
converges absolutely and, consequently, tends to zero as \(k \to \infty\).
Thus, if conditions (1)—(4) are satisfied, then
\[ R(P, P')=\frac{e^{2ikc}}{4ikc}\left[\int_l q(P'')\,dl+o(1)\right], \qquad k\to\infty, \tag{16} \]
where \(l\) coincides with the segment \(PP'\).
Let us note that condition (3) ensures the absolute convergence of the integral (15) as \(\rho \to \infty\), while condition 4 ensures it as \(\rho \to c\). Condition 4 admits a substantial weakening.
Let us also note that, imposing additional requirements on \(q(x,y,z)\) and using the properties of \(I(q,P,P',\rho)\) (item 1), one can find the subsequent terms of the expansion of \(R(P,P')\) in powers of \(1/k\) as \(k\to\infty\).
Consider the case, frequently encountered in applications, when \(q(x,y,z)\) has a discontinuity at the points of some bounded convex closed surface \(S\). The surface \(S\) divides all space into two nonintersecting parts \(R_1\) and \(R_2\), where \(R_1\) denotes the convex part of space. We shall say that the points \(P\) and \(P'\) form a focal pair of the surface \(S\) if the common points \(P''\) of the surface \(S\) and of some ellipsoid \(P''P+P''P'=2\rho\) form a set of positive planar measure. Denote by \(\chi(x,y,z)\) the characteristic function of \(R_1\).
Theorem 2. Let \(\Phi(x,y,z)=0\) be the equation of the surface \(S\), and let the function \(\Phi(x,y,z)\) have continuous first partial derivatives. Suppose, moreover, that \(q(P'')=q_1(P'')\chi(P'')\), where \(q_1(P'')\) has continuous first partial derivatives satisfying, for some \(\alpha\), the Hölder condition. If the points \(P\) and \(P'\) \((P,P'\in R_1)\) do not form a focal pair of the surface \(S\), then equality (16) holds.
In some cases the smoothness restrictions on \(q_1(P'')\) may be weakened. If the function \(q_1(P'')\) has the form \(q_1(P'')=q_1(r',r'')\), then for Theorem 2 to hold it is sufficient to require continuity of the partial derivatives
\[ \frac{\partial}{\partial r'} q_1(r',r'') \quad \text{and} \quad \frac{\partial}{\partial r''} q(r',r''). \]
Odessa Electrotechnical
Institute of Communications
Received
9 IX 1965
REFERENCES
- L. A. Sakhnovich, DAN, 168, No. 2 (1966).