UDC 517.9:539.1

MATHEMATICAL PHYSICS

A. G. RAMM

ON A REGION FREE OF RESONANCE POLES IN THE SCATTERING PROBLEM FOR A THREE-DIMENSIONAL POTENTIAL

(Presented by Academician V. A. Fock, 26 VI 1965)

In the study of the scattering problem for a spherically symmetric potential one is led to the study of the equation

\[ u''+\bigl(k^2 - V(r)\bigr)u=0, \tag{1} \]

assuming that the potential satisfies

\[ \int_0^\infty r\,|V(r)|\,dr<\infty . \tag{2} \]

An important question is the distribution of the complex eigenvalues of equation (1). These values correspond to quasistationary states of the particle. The constant \(k^2\) is proportional to the energy of the particle. The theory of quasistationary states is considered in papers \((^{1-3})\). In papers \((^4)\), under the assumption that the potential is finite, it is proved that the poles of the scattering matrix (which are also complex eigenvalues) for the one-dimensional equation (1) at high energies \(k^2 \gg 1\) are located in the lower half-plane of the complex variable \(k=\sigma+i\tau\) under the curve \(\tau=-\ln|\sigma|\). In the present paper an analogous result is obtained for the three-dimensional Schrödinger equation

\[ \Delta u+k^2u - V(x)u=0,\qquad x=(x_1,x_2,x_3), \tag{3} \]

under the condition that the potential is a differentiable finite function, and it is proved that the wave function \(u(x,k)\) admits analytic continuation into the entire lower half-plane \(\tau<0\), with its only singularities being poles. (It is known that for \(\tau>0\) the function \(u(x,k)\) is analytic in \(k\)*.) In paper \((^5)\) it is proved that, for the one-dimensional equation (1), finiteness of the potential is a necessary and sufficient condition for the existence, for the function \(u(x,k)\), of the above-mentioned analytic properties with respect to the variable \(k\). Questions of analytic continuation of the solution of the Schrödinger equation and of the Green’s function for the Schrödinger equation are the subject of papers \((^{6-9})\).

\(1^\circ.\) We shall prove that the solution of equation (3) admits analytic continuation into the half-plane \(\tau<0\), and that the only singularities of the analytic continuation are isolated poles \((^{6,7})\). The solution of equation (3) satisfies the integral equation

\[ u=e^{i(\bar{k},x)}+\int \frac{e^{ik|x-y|}}{-4\pi |x-y|}\,V(y)\,u(y,k)\,dy, \tag{4} \]

* If the operator (3) has no negative discrete spectrum. In the contrary case the function \(u(x,k)\) is not defined uniquely for those \(k=i\tau\), \(\tau>0\), for which \(k^2=-\tau^2\) coincides with one of the points of the discrete spectrum. The function \(u(x,k)\) may be chosen so that it is analytic for \(\tau>0\). In what follows, for simplicity, we shall assume that the discrete spectrum is absent.

where the integration is over the entire space, \(x=(x_1,x_2,x_3)\), \(y=(y_1,y_2,y_3)\) are points of three-dimensional space. It can be verified that the integral operator in (4) depends analytically on \(k\) and, for any complex \(k\), is a completely continuous operator in the space \(C(E_3,e^{-|x|})\). We shall now use the following theorem \((7,10)\), from which the required assertion follows.

Theorem 1. Let the operator \(T(\lambda)\) depend analytically on \(\lambda\), where \(\lambda \subset D\), and \(D\) is a connected domain in the plane of the complex variable \(\lambda\). Let \(T(\lambda)\) be a completely continuous operator for each \(\lambda \subset D\). Then either the operator \(I-T(\lambda)\) has no bounded inverse at any point \(\lambda \subset D\), or this inverse exists and is bounded for all \(\lambda \subset D\), except, possibly, for a countable number of isolated points.

\(2^0\). Let us establish that the poles of the function \(u(x,k)\), located in the half-plane \(\tau<0\), for large \(\sigma\) lie below the curve

\[ \tau=-a\ln|\sigma|+b, \tag{5} \]

where \(a>0\), \(b\) are constants. For the proof, note that the function \(u(x,k)\) has no singularities in a neighborhood of those points \(k\) for which equation (4) is uniquely solvable. We shall show that for points \(k\) lying above the curve (5), equation (4) is uniquely solvable as \(\sigma\to\infty\). This will prove the assertion made above. For the proof let us establish an estimate of the iterated kernel in equation (4):

\[ \mathcal{L}(x,y,k)\equiv \int \frac{e^{ik[|x-z|+|z-y|]}}{|x-z|\cdot |z-y|}\,V(z)\,dz; \qquad z=(z_1,z_2,z_3). \tag{6} \]

The estimate of the kernel \(\mathcal{L}(x,y,k)\) will be carried out according to the scheme proposed in \((11)\). Let us explain the further course of reasoning. We want to show that the kernel \(\mathcal{L}(x,y,k)\) will be small as \(k\to\infty\), if \(k\) lies above the curve (5). If this is proved, then, iterating equation (4) once, we arrive at an equation with a small kernel. This equation can be solved by the method of successive approximations, which will complete the proof. We proceed to the estimate of the kernel (6) for large complex \(k\). We introduce the change of variables used to obtain the estimate for real \(k\) in \((11)\), putting the coordinates of the point \(z\) equal to

\[ z_1=lst+(x_1-y_1)/2,\qquad z_2=l\sqrt{(s^2-1)(1-t^2)}\cos\psi+(x_2-y_2)/2, \tag{7} \]

\[ z_3=l\sqrt{(s^2-1)(1-t^2)}\sin\psi+(x_3-y_3)/2, \tag{8} \]

where \(\psi\) is the angle between the planes containing the vectors \(x-z\) and \(z-y\), and a fixed plane containing the vectors \(x\) and \(y\).

The Jacobian of the transformation is equal to

\[ J=l^3(s^2-t^2), \tag{9} \]

\[ 2l=|x-y|,\qquad |x-z|+|z-y|=2ls,\qquad |x-z|-|z-y|=2lt. \tag{10} \]

The integral (6) is transformed into the form:

\[ \mathcal{L} = l\int_0^{2\pi}d\psi \int_{-1}^{1}dt \int_1^\infty e^{2ikls}\,\widetilde V(s,t,\psi)\,ds. \tag{11} \]

Denote

\[ p(s)=\int_0^{2\pi}d\psi\int_{-1}^{1}\widetilde V(s,t,\psi)\,dt. \tag{12} \]

Then

\[ \mathcal{L}=l\int_1^\infty e^{2ikls}p(s)\,ds. \tag{13} \]

Together with the potential \(V\), the function \(p(s)\) is finite and differentiable. We have

\[ \mathcal L=-\frac{e^{2ikl}}{2ik}p(1)-\frac{1}{2ik}\int_1^\infty e^{2ikls}p'(s)\,ds\equiv \mathcal L_1+\mathcal L_2, \tag{14} \]

where the finiteness of \(p(s)\) was used. Let \(\tau=-\varphi(\sigma)\), \(\varphi(\sigma)>0\). The first term on the right-hand side of (14), as \(\sigma\to\infty\), has order

\[ \mathcal L_1=O\left(e^{2l\varphi(\sigma)}/\sqrt{\sigma^2+\varphi^2(\sigma)}\right), \tag{15} \]

the second term has order

\[ \mathcal L_2=o(\mathcal L_1). \tag{16} \]

Consequently,

\[ \mathcal L=O\left(e^{2l\varphi(\sigma)}/\sqrt{\sigma^2+\varphi^2(\sigma)}\right). \tag{17} \]

Choose the function \(\varphi(\sigma)\) so that, as \(\sigma\to\infty\),

\[ \mathcal L=o(1). \tag{18} \]

It is clear that in order for (18) to hold, one must set either

\[ \varphi(\sigma)=o(\ln|\sigma|), \tag{19} \]

or

\[ \varphi(\sigma)\le c\ln|\sigma|, \tag{20} \]

where \(c>0\) is a sufficiently small constant depending on \(l\).

Recall now that \(\tau=-\varphi(\sigma)\). Inequality (20) allows one to assert that the solution of equation (3) admits an analytic continuation into the domain \(\tau\ge -c\ln|\sigma|\). Our assertion is proved. It remains to note that the poles of the scattering amplitude

\[ f(\mathbf n,\vec\nu,k)=\int e^{-ik(\mathbf n,\mathbf y)}V(y)u(y,k)\,dy \tag{21} \]

coincide with the poles of the wave function \(u(y,k)\).

Thus, the following has been proved.

Theorem 2. If the potential is a finite differentiable function, then the solution \(u(x,k)\) of Schrödinger’s equation (3) admits an analytic continuation to the entire plane of the complex variable \(k\) as a meromorphic function. This continuation is regular in the half-plane \(\operatorname{Im} k\ge 0\). The poles of the function \(u(x,k)\) lie in the half-plane \(\operatorname{Im} k<0\) below the curve \(\tau=-a\ln|\sigma|+b\), where \(a>0\) and \(b\) are some constants. These assertions remain valid for the scattering amplitude.

3°. If, instead of finiteness of the potential, one assumes that the condition

\[ \int e^{-a|x|}|V(x)|\,dx<C(a),\qquad C(a)=\mathrm{const}, \tag{22} \]

is satisfied, where \(a>0\) is an arbitrary number, then the function \(u(x,k)\) will admit an analytic continuation to the entire plane of the complex variable \(k\) as a meromorphic function, regular in the half-plane \(\tau>-\tau_0\), where \(\tau_0>0\) is some constant. If condition (22) is satisfied for some fixed \(a>0\), then analytic continuation is possible, generally speaking, only into the half-plane \(\tau>-a/2\). It is meromorphic in this half-plane and regular for \(\tau>-\tau_0\). If the Schrödinger operator is considered in the exterior of a bounded domain with a Lyapunov boundary, and if some self-adjoint boundary condition is imposed on the boundary—

* See the footnote on p. 1319.
** In works \((^6,^7)\), in analogous assertions, \(\operatorname{Re}p>-a\) was erroneously written instead of \(\operatorname{Re}p>-a/2\).

condition, then the solution of the Schrödinger equation admits analytic continuation as a meromorphic function to the whole plane if the potential is finite, or satisfies condition (22), and to the half-plane \(\tau>-a/2\) if condition (22) is fulfilled for some fixed \(a\). In the case of a domain with a boundary, we are unable to indicate a domain of regularity of the analytic continuation in the half-plane \(\tau<0\). In the half-plane \(\tau>0\) the analytic continuation is regular*. In the planar case the assertions made above remain valid after the following changes are introduced. In all assertions the analytic continuation is performed to the plane with a cut along the negative imaginary semiaxis. The assertions made can be applied to estimating the rate at which the solution of a nonstationary problem tends to the limiting amplitude. Let us note here only the following fact. In a planar (or even-dimensional) space the presence of a cut under analytic continuation entails the phenomenon of wave diffusion. If in three-dimensional space, for finite and smooth potentials and initial data, the solution of the problem

\[ u_{tt}+\mathcal L u=0,\qquad u|_{t=0}=0,\qquad u_t|_{t=0}=g(x), \tag{23} \]

where

\[ \mathcal L u=-\Delta u+V(x)u \tag{24} \]

decreases exponentially as \(t\to\infty\), then in the two-dimensional case, generally speaking, it decreases no faster than \(O(1/t)\), even for finite and infinitely differentiable functions \(V(x)\) and \(g(x)\).

Received
24 VI 1965

CITED LITERATURE

\({}^{1}\) N. S. Krylov, V. A. Fok, ZhETF, 17, 93 (1947).
\({}^{2}\) L. A. Khalfin, DAN, 115, No. 2, 277 (1957).
\({}^{3}\) Ya. B. Zel’dovich, ZhETF, 39, No. 9, 776 (1960).
\({}^{4}\) T. Regge, Collection “Mathematics,” 7, 4, 836 (1963).
\({}^{5}\) A. G. Ramm, DAN, 157, No. 5, 1073 (1964).
\({}^{6}\) A. G. Ramm, UMN, 19, No. 5, 192 (1964).
\({}^{7}\) A. G. Ramm, Dokl. AN AzerbSSR, 21, No. 1, 3 (1965).
\({}^{8}\) A. G. Ramm, Abstracts of Reports at the Third All-Union Symposium on Wave Diffraction, “Nauka,” 1964, p. 28.
\({}^{9}\) A. G. Ramm, Izv. Vyssh. Uchebn. Zaved., Mathematics, 2, 136 (1965).
\({}^{10}\) N. Dunford, J. Schwartz, Linear Operators, 1, IL, 1962, p. 636.
\({}^{11}\) L. D. Faddeev, Vestn. Leningrad Univ., 7, 2, 126 (1956).

* See footnote on p. 1321.
** See footnote on p. 1319.