Full Text
UDC 517.946.4
PROPERTIES OF INTEGRALS ALONG A PARABOLA AND BORN APPROXIMATIONS OF SCATTERING THEORY
I. M. POLISHCHUK
1. In [1] the properties of integrals over a paraboloid were studied, and on this basis the asymptotics of the Born approximations of scattering theory for the three-dimensional Schrödinger equation were obtained. Below an analogous study is carried out for the two-dimensional case. This case is quite close to certain scalar problems of electrodynamics. A number of proofs analogous to those omitted in [1] are given below.
Thus, consider the two-dimensional Schrödinger equation
\[ \Delta_{x,y}\psi+\bigl[k^2-q(x,y)\bigr]\psi=0 . \tag{1} \]
(\(k\) is assumed real; the requirements on the function \(q\), which may be complex, will be clear from what follows (see Theorem 2).)
We seek in the whole space a continuous solution satisfying the condition
\[ \psi-e^{ikx}\to 0,\qquad r\to\infty . \]
As is known, the formal series corresponding to the method of successive approximations has the form
\[ \psi=\sum_{n=0}^{\infty}\psi_n, \tag{2} \]
where \(\psi_0=e^{ikx}\), \(\psi_n=A^n\psi_0\),
\[ A=\frac{i}{4}\int q(x',y')H_0^{(1)}(kr')\,dS', \]
the integral is taken over the whole plane,
\[ r'=\sqrt{(x-x')^2+(y-y')^2},\qquad dS'=dx'\,dy' . \]
Below we study the dependence of \(\psi_1\) on \(q\) and \(k\) as \(k\to\infty\), both for a smooth function \(q(x,y)\) and for a function \(q(x,y)\) having a jump on a smooth curve. In the latter case, points called foci are identified, at which the asymptotic expansions are different from those at ordinary points.
Let us write the correction \(\psi_1\) of the first Born approximation to the wave function:
\[ \psi_1(x,y)=\frac{i}{4}e^{ikx}\int e^{iks}q(x+s,y+t)H_0^{(1)}(kr)\,dS, \tag{3} \]
where \(r^2=s^2+t^2\), \(dS=ds\,dt\).
Since we are interested in the behavior of \(\psi_1\) as \(k\to\infty\), we shall assume \(k\) so large that in expression (3) one may use the asymptotic formula for \(H_0^{(1)}(x)\) for \(x\gg 1\). This possibility follows from the uniform convergence of the integral in (3).
Indeed, represent (3) in the form
\[ \psi_1=F_1+F_2, \tag{4} \]
where
\[ F_1(x,y)=\frac{i}{4}e^{ikx}\int_{S_\varepsilon} e^{iks}q(x+s,y+t)H_0^{(1)}(kr)\,dS, \]
\[ F_2(x,y)=\frac{i}{4}e^{ikx}\int_{S_\infty-S_\varepsilon} e^{iks}q(x+s,y+t)H_0^{(1)}(kr)\,dS, \]
\(S_\infty\) is the whole plane, and \(S_\varepsilon\) is the circular domain with center at the origin, determined by the condition
\[ kr\leq N,\qquad \text{where } N\gg 1. \tag{5} \]
Inequality (5) permits us, in the second term of formula (4), to use the asymptotic expression for \(H_0^{(1)}(x)\) for \(x\gg 1\). On the other hand, the fulfillment of (5) leads to the fact that the principal term of the asymptotics of \(\psi_1\) as \(k\to\infty\) is determined by the second term of formula (4).
In fact, if one takes into account that
\[ |H_0^{(1)}(x)|<\frac{C}{\sqrt{x}}\quad \text{for all }x, \]
\[ H_0^{(1)}(x)=\sqrt{\frac{2}{\pi x}}\,e^{i\left(x-\frac{\pi}{4}\right)} +O\left(x^{-\frac{3}{2}}\right),\qquad x\to\infty, \]
then for a bounded function \(q\) (\(|q(x,y)|<M\)), different from zero in a finite part of the plane, we shall have
\[ |F_1(x,y)|\leq \frac{\pi C M N^{\frac{3}{2}}}{3k^2}, \tag{6} \]
\[ F_2(x,y)=\frac{e^{i\frac{\pi}{4}}}{2\sqrt{2\pi}}\, \frac{e^{ikx}}{\sqrt{k}} \int_{S_\infty-S_\varepsilon} e^{ik(s+r)}q(x+s,y+t)\frac{dS}{\sqrt{r}} +O\left(k^{-\frac{3}{2}}\right). \]
From (4), (6), and the relation following from (5),
\[ S_\varepsilon\to 0,\qquad k\to\infty, \]
there follows the validity of what is being proved. (The proof is entirely analogous also in the case when \(q\) satisfies the conditions of Theorem 2.) Thus, we have
\[ \psi_1=\tilde{\psi}_1+O\left(k^{-\frac{3}{2}}\right), \tag{7} \]
where
\[ \tilde{\psi}_1(x,y)= \frac{e^{i\frac{\pi}{4}}}{2\sqrt{2\pi}}\, \frac{e^{ikx}}{\sqrt{k}} \int_{S_\infty} e^{ik(s+r)}q(x+s,y+t)\frac{dS}{\sqrt{r}}. \]
The expression \(\tilde{\psi}_1\), generally speaking, cannot be investigated by the method of stationary phase, since the points at which the phase is stationary fill the entire negative half-axis of the \(s\)-axis.
As we shall see below, the principal part of \(\psi_1(x,y)\) is determined by an integral along this “special” line.
Along with the Cartesian coordinates \((s,t)\) of a point, let us also introduce the coordinates \((\rho,\varphi)\) and \((\rho,s)\), where \(\rho=r+s\), and \(\varphi\) is the polar angle with respect to the point of observation \(P(x,y)\). Then
\[ \widetilde{\psi}_1(x,y)= \frac{e^{i\frac{\pi}{4}}}{2\sqrt{2\pi}}\, \frac{e^{ikx}}{\sqrt{k}} \int_{0}^{\infty} e^{ik\rho} J(q,\rho,P)\,d\rho . \tag{8} \]
Depending on the coordinate system, \(J(q,\rho,P)\) is defined by the formulas
\[ J(q,\rho,P)= \int_{-\pi}^{\pi} \frac{\sqrt{\rho}}{(1+\cos\varphi)^{3/2}}\, q\!\left(x+\frac{\rho\cos\varphi}{1+\cos\varphi}, y+\frac{\rho\sin\varphi}{1+\cos\varphi}\right)d\varphi , \tag{9} \]
\[ J(q,\rho,P)= \int_{-\infty}^{\rho/2} \sqrt{\frac{\rho-s}{\rho^2-2\rho s}}\, \left[ q\!\left(x+s,\;y+\sqrt{\rho^2-2\rho s}\right)+ q\!\left(x+s,\;y-\sqrt{\rho^2-2\rho s}\right) \right]ds . \tag{10} \]
The expressions \(J(q,\rho,P)\) are integrals of the function \(q\) along the parabola with focus at the point \(P\).
- In the present subsection we shall derive the basic properties of the integral along a parabola, thus transferring some results of [1] to the two-dimensional case.
Notation: \(J(q,\rho)=J(q,\rho,P)\), if \(x=y=0\).
1) If \(q(s,t)\) and
\[ f=\frac{1}{2}q+s\frac{\partial q}{\partial s} +t\frac{\partial q}{\partial t} \]
are integrable along the parabola, then, as follows from (9),
\[ \frac{d}{d\rho}J(q,\rho)=\frac{1}{\rho}J(f,\rho), \qquad \rho>0. \]
2) Let
\[ q(s,t)=a(s)t^n, \]
where
\[ \sqrt{\frac{\rho-s}{\rho^2-2\rho s}}\,a(s)s^k \in L\!\left(-\infty,\frac{\rho}{2}\right), \qquad 0\le k\le \frac{n}{2}, \]
and \(n\) is a nonnegative integer. Then the equality
\[ J(q,\rho)=0, \tag{11} \]
holds if \(n\) is odd. If \(n\) is even, then
\[ J(q,\rho)= 2\int_{-\infty}^{\rho/2} \sqrt{\rho-s}\,(\rho^2-2\rho s)^{\frac{n-1}{2}}a(s)\,ds . \tag{12} \]
Equalities (11) and (12) make it possible to compute the integral over a parabola of a function \(q(s,t)\) that is analytic in the variable \(t\).
3) For \(q(s,t)\) of the form
\[ q(s,t)=\sum_{k\geqslant 0,\; k<n} a_k(s)t^k\,\frac{1}{k!}+q_n(s,t), \]
where
\[ a_k(s)=\left.\frac{\partial^k q}{\partial t^k}\right|_{t=0},\qquad \sqrt{\frac{\rho-s}{\rho^2-2\rho s}}\,a_k(s)s^m\in L\left(-\infty,\frac{\rho}{2}\right), \qquad 0\leqslant m\leqslant \frac{k}{2}, \]
\[ \lim_{n\to\infty}J(q_n,\rho)=0, \]
the equality holds
\[ J(q,\rho)=2\sum_{k=0}^{\infty}\frac{1}{(2k)!} \int_{-\infty}^{\rho/2} \sqrt{\rho-s}\,(\rho^2-2\rho s)^{k-\frac{1}{2}}\,a_{2k}(s)\,ds. \]
4) If \(q(s,t)\) is continuous and satisfies the inequality
\[ \left|q\left(s,\pm\sqrt{\rho^2-2\rho s}\right)\right|\leqslant \varphi(s),\qquad s\leqslant 0, \]
where \(\varphi(s)\in L(-\infty,0)\), then
\[ J(q,\rho)=\sqrt{\frac{2}{\rho}}\int_{-\infty}^{0}q(s,0)\,ds+O\left(\rho^{\frac{1}{2}}\right),\qquad \rho\to 0. \tag{13} \]
Indeed,
\[ J(q,\rho)= \int_{-\infty}^{-\rho N} \sqrt{\frac{\rho-s}{\rho^2-2\rho s}} \left[ q\left(s,\sqrt{\rho^2-2\rho s}\right) + q\left(s,-\sqrt{\rho^2-2\rho s}\right) \right]\,ds+ \]
\[ + \int_{-\rho N}^{\rho/2} \sqrt{\frac{\rho-s}{\rho^2-2\rho s}} \left[ q\left(s,\sqrt{\rho^2-2\rho s}\right) + q\left(s,-\sqrt{\rho^2-2\rho s}\right) \right]\,ds. \tag{14} \]
Taking \(N\) large, in the first term we have
\[ \sqrt{\frac{\rho-s}{\rho^2-2\rho s}}\sim \frac{1}{\sqrt{2\rho}},\qquad \rho\to 0, \]
and therefore
\[ \int_{-\infty}^{-\rho N} \sqrt{\frac{\rho-s}{\rho^2-2\rho s}} \left[ q\left(s,\sqrt{\rho^2-2\rho s}\right) + q\left(s,-\sqrt{\rho^2-2\rho s}\right) \right]\,ds \sim \]
\[ \sim \sqrt{\frac{2}{\rho}}\int_{-\infty}^{0}q(s,0)\,ds,\qquad \rho\to 0. \]
Estimating, as \(\rho\to 0\), the magnitude of the second term in formula (14), we are convinced of the validity of formula (13).
5) From the inequality
\[ |q(s,t)|\leqslant \varphi(r) \]
follows that
\[ |J(q,\rho)| \leqslant 2 \int_{\rho/2}^{\infty} \sqrt{\frac{r}{2\rho r-\rho^2}}\,\varphi(r)\,dr . \tag{15} \]
Inequality (15) makes it possible to judge the behavior of \(J(q,\rho)\) as \(\rho \to \infty\).
- In this subsection we investigate the integral over a parabola of a function that is zero outside a smooth curve.
According to property 1 of subsection 2, for smooth functions \(q(s,t)\) there exists
\[ \frac{d}{d\rho} J(q,\rho), \quad \rho>0. \]
We shall now determine the conditions for existence and a method of computing
\[ \frac{d}{d\rho} J(q,\rho) \]
in the case when \(q(s,t)\) has a discontinuity on some curve. A potential of this type often occurs in applications.
Fig. 1
Fig. 2
Let \(\Phi(s,t)=0\) be the equation of a curve \(L\), which divides the whole plane into two nonintersecting parts \(R_1\) and \(R_2\). We shall assume, moreover, that \(\Phi(s,t)\) has continuous first-order partial derivatives.
Following [1], define on \(L\) a function \(\varepsilon(s,t)\) as follows: \(\varepsilon(s,t)=-1\) if the branch of the parabola \(s+r=\rho\) enters \(R_1\) through the point \(Q(s,t)\) of the curve \(L\), and \(\varepsilon(s,t)=+1\) in the case when the branch of the parabola exits \(R_1\) (Fig. 1).
Denote by \(\chi(s,t)\) the characteristic function of the closure of \(R_1\), and by \(\gamma_\rho\) the set of points of intersection of the line \(L\) with the parabola \(\rho=s+r\). For the coordinates of the intersection points lying in the upper and lower half-planes, we introduce respectively the notation \((s^+,t^+)\), \((s^-,t^-)\).
Next define on \(\gamma_\rho\) the following function:
\[ \Delta(s,t)=\frac{\partial \Phi}{\partial s}(\rho-2s)-t\frac{\partial \Phi}{\partial t}. \tag{16} \]
The equality \(\Delta(s,t)=0\) at some point \(M(s,t)\in\gamma_\rho\) means that the tangent to the parabola at the point \(M\) coincides with the tangent to \(L\) at the same point.
Theorem 1. Suppose that at the points of \(\gamma_\rho\) the inequality
\[ \Delta(s,t)\geqslant \delta>0 \]
holds. If the functions \(q\chi\) and \(f\chi=\left(\dfrac{1}{2}q+s\dfrac{\partial q}{\partial s}+t\dfrac{\partial q}{\partial t}\right)\chi\) are integrable over the parabola and \(q\) is bounded at the points belonging to \(\gamma_\rho\), then the relation
\[ \frac{d}{d\rho} J(q\chi,\rho)=\frac{1}{\rho}J(f\chi,\rho)+\frac{1}{\rho} \sum_{(s_i,t_i)\in\gamma_\rho} \sqrt{\frac{\rho-s_i}{\rho^2-2\rho s_i}} \left(s_i\frac{\partial\Phi(s_i,t_i)}{\partial s}+ t_i\frac{\partial\Phi(s_i,t_i)}{\partial t}\right) q(s_i,t_i)\varepsilon(s_i,t_i)\frac{\rho-2s_i}{\Delta(s_i,t_i)} . \]
Proof. We first consider the simplest case, when \(L\) is a straight line whose equation is
\(s\cos\alpha+t\cos\beta-p=0\). For definiteness we shall study the situation depicted in Fig. 2.
Introduce the auxiliary function
\[ \chi_\delta= \begin{cases} 1, & \text{if } s\cos\alpha+t\cos\beta-p\le -\delta,\\[4pt] -\dfrac{s\cos\alpha+t\cos\beta-p}{\delta}, & \text{if } -\delta<s\cos\alpha+t\cos\beta-p\le 0,\\[8pt] 0, & \text{if } s\cos\alpha+t\cos\beta-p>0. \end{cases} \]
It is clear that
\[ \lim_{\delta\to 0}\chi_\delta=\chi . \tag{17} \]
Applying property 1 of item 2 to the function \(q\chi_\delta\), we obtain
\[ \frac{d}{d\rho}J(q\chi_\delta,\rho)= \frac{1}{\rho}J\left[\left(\frac12 q+s\frac{\partial q}{\partial s} +t\frac{\partial q}{\partial t}\right)\chi_\delta,\rho\right] +\frac{1}{\rho}J(q\vartheta_\delta,\rho), \tag{18} \]
where
\[ \vartheta_\delta= \begin{cases} 0, & \text{if } s\cos\alpha+t\cos\beta-p\le -\delta,\\[4pt] -\dfrac{s\cos\alpha+t\cos\beta}{\delta}, & \text{if } -\delta<s\cos\alpha+t\cos\beta-p\le 0,\\[8pt] 0, & \text{if } s\cos\alpha+t\cos\beta-p>0. \end{cases} \]
Using formula (10), we obtain
\[ \begin{aligned} J(q\vartheta_\delta,\rho) ={}&-\frac{1}{\delta}\int_{s_1^+}^{s_2^+} \sqrt{\frac{\rho-s}{\rho^2-2\rho s}}\, q\left(s,\sqrt{\rho^2-2\rho s}\right) \left(s\cos\alpha+\sqrt{\rho^2-2\rho s}\cos\beta\right)\,ds\\ &-\frac{1}{\delta}\int_{s_1^-}^{s_2^-} \sqrt{\frac{\rho-s}{\rho^2-2\rho s}}\, q\left(s,-\sqrt{\rho^2-2\rho s}\right) \left(s\cos\alpha-\sqrt{\rho^2-2\rho s}\cos\beta\right)\,ds, \end{aligned} \tag{19} \]
where \(s_1^\pm\) and \(s_2^\pm\) are the abscissas of the points of intersection of the parabola
\(s+r=\rho\), respectively, with the straight lines
\(s\cos\alpha+t\cos\beta-p=0\) and
\(s\cos\alpha+t\cos\beta-p=-\delta\).
It is easy to see that
\[ s_2^\pm-s_1^\pm= -\frac{\delta}{\cos\alpha\mp \dfrac{\rho}{\sqrt{\rho^2-2\rho \xi^\pm}}\cos\beta}, \tag{20} \]
where \(s_1^\pm<\xi^\pm<s_2^\pm\).
Using (19) and (20), we obtain
\[ \begin{aligned} \lim_{\delta\to 0} J(q_{\gamma_\delta}, \rho) &= \sqrt{\frac{\rho-s^{+}}{\rho^{2}-2\rho s^{+}}}\, q(s^{+},t^{+})(s^{+}\cos\alpha+t^{+}\cos\beta)\times \\ &\quad \times \frac{\rho-2s^{+}}{\cos\alpha(\rho-2s^{+})-t^{+}\cos\beta} + \\ &\quad + \sqrt{\frac{\rho-s^{-}}{\rho^{2}-2\rho s^{-}}}\, q(s^{-},t^{-})(s^{-}\cos\alpha+t^{-}\cos\beta)\times \\ &\quad \times \frac{\rho-2s^{-}}{\cos\alpha(\rho-2s^{-})-t^{-}\cos\beta}. \end{aligned} \tag{21} \]
From formulas (18), (17), and (21) it follows that the theorem is valid for the particular case considered. Carrying out an analogous procedure for different positions of the line, it is not difficult to verify the validity of the theorem for any line. Moreover, replacing on small intervals (near the points of intersection) the line \(L\) by the corresponding tangent lines, we obtain a proof of the theorem in the general case.
- In this section we shall find the asymptotics of the integral
\[ \int_{0}^{\infty} e^{ik\rho} J(q,\rho,P)\,d\rho,\qquad k\to\infty . \tag{22} \]
The following holds.
Theorem 2. If
\[ |q(x+s,y+t)|<\varphi_{1}(r,P),\qquad |f(P,Q)|<\varphi_{2}(r,P), \]
where
\[ f(P,Q)=-\frac12\,q(x+s,y+t)+ \left(s\frac{\partial}{\partial s}+t\frac{\partial}{\partial t}\right) q(x+s,y+t), \]
\[ \int_{a}^{\infty}\sqrt{\frac{r}{r-a}}\,\varphi_{1,2}(r,P)\,dr<\infty, \]
\(a\) is any positive number, then the formula
\[ \int_{0}^{\infty} e^{ik\rho} J(q,\rho,P)\,d\rho = \frac{\sqrt{2\pi}\,e^{-i\frac{3}{4}\pi}}{\sqrt{k}} \int_{-\infty}^{x} q(s,y)\,ds +O(k^{-1}) \tag{23} \]
is valid.
Indeed, it is easy to see that, under the adopted assumptions, expression (22) can be represented in the form
\[ \int_{0}^{\infty} e^{ik\rho}J(q,\rho,P)\,d\rho = \int_{0}^{\delta} e^{ik\rho}J(q,\rho,P)\,d\rho - \frac{e^{ik\delta}}{ik}J(q,\delta,P) - \]
\[ - \frac{1}{ik}\int_{\delta}^{\infty} e^{ik\rho}\,\frac{1}{\rho}\,J(f,\rho,P)\,d\rho . \tag{24} \]
Taking \(\delta\) small, we find, using Property 4 of § 2 and some known results of the theory of asymptotic expansions [2]:
\[ \int_0^\delta e^{ik\rho} J(q,\rho,P)\,d\rho = \frac{\sqrt{2\pi}\,e^{-i\frac{3}{4}\pi}}{\sqrt{k}} \int_{-\infty}^x q(s,y)\,ds + O(k^{-1}). \]
By virtue of the assumptions made, the function \(q\) is integrable along the parabola and, consequently, the second term in (24) is asymptotically small in comparison with the first. It is not difficult to see that the third term in (24) is also asymptotically small in comparison with the first. Therefore formula (23) is valid. From formulas (7), (8), and (23) we conclude that
\[ \psi_1(x,y) = \frac{e^{ikx}}{2ik} \int_{-\infty}^x q(s,y)\,ds + O\!\left(k^{-\frac{3}{2}}\right). \tag{25} \]
A nonrigorous derivation of formula (25) is known (see, for example, [3]).
Let us now consider a bounded closed curve \(L\), which divides the entire plane into two nonintersecting parts \(R_1\) and \(R_2\) and has equation \(\Phi(x,y)=0\). It is assumed that \(\Phi(x,y)\) has continuous first-order partial derivatives. For simplicity we shall take the curve \(L\) to be convex.
We shall say that \(P(x,y)\) is a point of focus type of the curve \(L\) if the common points of the curve \(L\) and of some parabola
\[
(y'-y)^2=\rho^2-2\rho(x'-x), \qquad \rho>0,
\]
form a set of positive linear measure.
Let the function \(q_1(x,y)\) have continuous first-order partial derivatives. Put, as was already done in Sec. 3,
\[ q(x,y)=q_1(x,y)\chi(x,y), \]
where \(\chi(x,y)\) is the characteristic function of the closure \(R_1\).
If \(P(x,y)\in R_1\) and \(P\) is not a point of focus type of the curve \(L\), then as \(k\to\infty\)
\[ \psi_1(x,y) = \frac{e^{ikx}}{2ik} \int_{x_0}^x q(s,y)\,ds + O\!\left(k^{-\frac{3}{2}}\right), \]
where \(\Phi(x_0,y)=0,\quad x_0<x\).
Now let \(P\in R_2\), not be a point of focus type, and be located in the “shadow,” i.e. \(\Phi(x,y)=0\) has two simple roots \(x_1\) and \(x_2\), with (since \(P(x,y)\) is located in the “shadow”) \(x_1<x_2<x\). Then, as \(k\to\infty\), we have
\[ \psi_1(x,y) = \frac{e^{ikx}}{2ik} \int_{x_1}^{x_2} q(s,y)\,ds + O\!\left(k^{-\frac{3}{2}}\right). \]
The case where \(P\in R_2\) is located in the illuminated region is investigated by means of the method of stationary phase.
In conclusion, the author considers it his duty to express gratitude to L. A. Sakhnovich for posing the problem and for his attention to the work.
References
- Sakhnovich L. A. Dokl. Akad. Nauk SSSR, 168, No. 2, 288–291, 1966.
- Erdélyi A. Asymptotic Expansions. Moscow, GIFML, 1963.
- Landau L. D., Lifshitz E. M. Quantum Mechanics. Moscow, GIFML, 1963.
Received by the editors
May 4, 1966
Odessa Electrotechnical
Institute of Communications