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UDC 517.946.4
MATHEMATICS
A. A. ARSEN’EV
SCATTERING OF A PLANE WAVE BY A SINGULAR POTENTIAL
(Presented by Academician A. N. Tikhonov on 28 IV 1967)
Numerous works have been devoted to quantum scattering theory in the stationary formulation; closest to us in subject matter are the works \((^{1,2})\). In all the works the potential \(V(x)\) was assumed to be an integrable function, since otherwise the Lippmann—Schwinger equation, which served as the principal tool of investigation in \((^{1,2})\), has an essential singularity (some special cases of singularities were studied by L. G. Mikhailov \((^3)\)). However, from the physical point of view the assumption of integrability of the potential \(V(x)\) is too restrictive: there exist problems in which the potential \(V(x)\) becomes \(+\infty\) on a set of positive measure. We propose such an integral equation whose kernel is a bounded integrable function independently of the dimension of the space \(N\) and of the singularity of the potential \(V(x)\).
Let \(R_N\) be \(N\)-dimensional Euclidean space \((N \geqslant 3)\); \(\Delta\) the Laplace operator; \(V(x)\) a scalar function (potential), \(0 \leqslant V(x) \leqslant \infty\); \(\Omega = \{x;\, V(x)=+\infty\}\). \(\widetilde H\) is the operator defined on functions finite in \(R_N \setminus \Omega\) by the formula \(\widetilde H u = -\Delta u + V(x)u\).
Put \(V_M(x)=\min\{V(x),M\}\) \((M \geqslant 0)\), and in what follows we shall mark by the index \(M\) all quantities referring to the potential \(V_M(x)\).
Below we assume everywhere that the potential \(V(x)\) satisfies the following two conditions: 1) every function \(V_M(x)\) is nonnegative and satisfies the Hölder condition (locally); 2) there exist constants \(a>0\), \(C<\infty\), \(R<\infty\), independent of \(M\), such that for all \(x\) lying outside the ball of radius \(R\), the estimate \(V(x)\leqslant C|X|^{-N-a}\) holds.
Consider the Cauchy problem for the heat equation (in the whole space):
\[ u(x,0)=u_0(x),\qquad \partial u/\partial t=-\widetilde H_M u,\qquad x\in R_N,\quad t>0,\quad u\in L^\infty, \tag{1} \]
and let \(G_M(x,y,t)\) be the Green function of problem (1). Represent it in the form:
\[ G_M(x,y,t)=G_0(x,y,t)-g_M(x,y,t); \]
\[ G_0(x,y,t)=(4\pi t)^{-N/2}\exp\{-(x-y)^2/4t\}. \]
Lemma 1. 1. For any \(t>0\) there exists the limit
\[ \lim_{M\to\infty} g_M(x,y,t)=g(x,y,t). \]
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In the uniform operator topology of the space \([L^p\to L^q;\; 1\leqslant p\leqslant\infty,\; 1\leqslant q<\infty]\) the operators \(\hat g_M(t)\)* converge to the operator \(\hat g(t)\).
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The operator \(\hat g(t)\) is completely continuous as an operator from \([L^p\to L^q,\; 1<p\leqslant\infty,\; 1\leqslant q<\infty]\).
* \([A_0\to B,\pi]\) is the space of all bounded linear operators from a Banach space \(A\) into a Banach space \(B\), where \(A\) and \(B\) satisfy condition \(\pi\).
** \(\hat g(t)\) is the integral operator with kernel \(g(x,y,t)\).
Corollary. The operators \(\hat G_M(t)\) converge to the operator \(\hat G(t)\); the operator function \(\hat G(t)\) is a semigroup in \(L^p\), \(1 < p < \infty\).
It can be verified that in \(L^p(R_N \setminus \Omega)\) the operator function \(G(t)\) is a semigroup of class \(C_0\).
Convergence of the operators \(\hat G_M(t)\) to \(\hat G(t)\) in the metric \([L^\infty \to L^\infty]\) need not hold; however, \(\hat G(t)\) is a semigroup in \(L^\infty\). Let \(\hat A\) be the infinitesimal operator of the semigroup \(\hat G(t)\), \(Hu = -\hat A u\). The operator \(H\) is an extension of the operator \(\hat H\).
The problem that we have to solve is formulated as follows: to find that solution of the equation
\[ Hu = \lambda u, \qquad x \in R_N, \qquad u \in L^\infty, \qquad \lambda > 0, \tag{2} \]
which can be represented in the form
\[ u(x,k) = e^{ikx} + \varphi(x,k), \tag{3} \]
where \(k^2=\lambda\), and the function \(\varphi(x,k)\), as \(|x|\to\infty\), satisfies the estimates
\[ \varphi(x,k)=O\left(|x|^{(1-N)/2}\right) \left(\frac{\partial}{\partial |x|}- i|k|\right)\varphi(x,k) o\left(|x|^{(1-N)/2}\right), \quad |x|\to\infty . \tag{4} \]
Consider the equation
\[ e^{-\lambda t}u=\hat G(t)u . \tag{5} \]
Substituting (3) into (5), we obtain that the function \(\varphi(x,k)\) satisfies the equation
\[ (e^{-\lambda t}-\hat G_0)\varphi=\hat g(t)\varphi . \]
Let \(K^+(\lambda)\) be the integral operator with kernel
\[ K^+(\lambda,r_{xy})= \left(\frac{1}{2\pi}\right)^{N/2} \lim_{\varepsilon\to+0} \int_0^\infty \frac{e^{-\rho^2 t}} {e^{-(\lambda+i\varepsilon)t}-e^{-\rho^2 t}} \frac{J_{N/2\alpha-1}(r_{xy}\rho)} {(r_{xy})^{N/2\alpha-1}} \rho^{N/2}\,d\rho \]
\[ (r_{xy}=|x-y|). \]
Lemma 2. If \(\varphi(x)\in L^p\cap L^\infty\), where \(p>2N/(N-1)\), then
\[ (e^{-\lambda t}-G_0)e^{\lambda t}(E+K^+(\lambda))\varphi = e^{\lambda t}(E+\hat K^+(\lambda))(e^{-\lambda t}-G_0)\varphi = \varphi . \]
Put, by definition,
\[ \hat T^+(\lambda)=-e^{\lambda t}(E+\hat K^+(\lambda))\hat g, \qquad \hat T_M^+(\lambda)=-e^{\lambda t}(E+\hat K^+(\lambda))\hat g_M \]
and consider the equations
\[ \psi(x,k,\lambda)= \bigl(\hat T^+(\lambda)(e^{iky}+\psi)\bigr)(x,k,\lambda), \qquad 0<\lambda<\infty,\qquad k\in R_N; \tag{6} \]
\[ \psi_M(x,k,\lambda)= \bigl(\hat T_M^+(\lambda)(e^{iky}+\psi_M)\bigr)(x,k,\lambda), \qquad 0<\lambda<\infty,\qquad k\in R_N. \tag{7} \]
The number \(t>0\) enters these equations as a parameter.
Let \(\Omega_1\) be the largest open connected set contained in the set \(R_N\setminus\Omega\) and containing points of the sphere \(|x|=2R\); \(\Omega_2=R_N\setminus(\Omega\cup\Omega_1)\).
The main result of our work is formulated in Theorems 1–3.
Theorem 1. There exists a countable set of points \(\{\lambda_i\}\), independent of \(t>0\), having the property that for each \(\lambda_i\) the equation
\[ H\varphi=\lambda_i\varphi \]
has \(m_i\), \(1\le m_i<\infty\), nontrivial solutions; all these solutions belong to \(L^\infty\), vanish outside the set \(\Omega_2\) (whence it follows that if \(\operatorname{mes}\Omega_2=0\), then the set \(\{\lambda_i\}\) is empty) and satisfy the equality \(T^+(\lambda_i)\varphi=\varphi\), while for any \(\lambda\in(0,\infty)\setminus\{\lambda_i\}\) the operator \((E-T^+(\lambda))^{-1}\in[L^q\to L^q,\; 2N/(N-1)<q\le\infty]\).
Theorem 2. The set of points \(\{\lambda_i\}\) discussed in Theorem 1 has the following properties: it is located on the positive ray from the point \(0\), and this distance is bounded below by a quantity depending only on the measure of the domain \(\Omega_2\); moreover, the number of points of the set \(\{\lambda_i\}\) lying in the interval \((0,\lambda)\) satisfies the asymptotic estimate
\[ N(\lambda)=\sum_{\lambda_i\leqslant \lambda}1\leqslant \frac{\operatorname{mes}\Omega_2}{(4\pi)^{N/2}\Gamma(N/2+1)}\lambda^{N/2}+o(\lambda^{N/2}). \]
Theorem 3. If \(\lambda\notin\{\lambda_i\}\), then there exists a solution of problem (2)—(4), namely
\[ \varphi(x,k)=\psi(x,k,k^2), \]
where \(\psi\) is a solution of equation (6). For sufficiently small \(t\), this solution exists, is unique, and is the limit as \(M\to\infty\), in the metric \(L^q\), \(2N/(N-1)<q<\infty\), of solutions of equation (7).
The author expresses his deep gratitude to A. A. Samarskii and V. A. Il’in for numerous consultations, and to the participants of A. N. Tikhonov’s seminar for discussion of the results.
Moscow State University
named after M. V. Lomonosov
Received
6 IV 1967
REFERENCES
- Teruo Ikebe, Arch. Rat. Mec. and Analysis, 5, No. 1, 1 (1960).
- A. Ya. Povzner, Matem. sborn., 32, (74), 1, 109 (1953).
- L. G. Mikhailov, Tr. AN TadzhSSR, Department of Phys. and Math., 1, Dushanbe, 1963.