UDC 517.9 : 539.1

MATHEMATICAL PHYSICS

A. G. RAMM

ON THE SPECTRUM OF THE SCHRÖDINGER OPERATOR USED IN THE OPTICAL MODEL OF THE NUCLEUS

(Presented by Academician V. A. Fock on 20 IX 1965)

In the problem of scattering of nucleons by nuclei \((^1)\), the Schrödinger equation is considered

\[ \mathcal{L}u - k^2u = -\Delta u - [k^2 - V(x) - T(r)(\bar l,\bar s)]u = 0,\qquad k^2>0, \tag{1} \]

\[ x=(x_1,x_2,x_3). \]

In this equation the wave function \(u\) is a two-component spinor, i.e., a quantity transformed according to the representation \(D_{1/2}\) of the rotation group of three-dimensional Euclidean space \((^2)\). The potential \(V(x)\) and the function \(T(r)\) are real functions tending to zero as \(r=|x|\to\infty\). As a rule, one restricts oneself to the case of a spherically symmetric potential, but we shall consider the general case*. The operator \(\bar l\) is the operator of orbital angular momentum. Its components have the form

\[ l_x=i(\sin\varphi\,\partial/\partial\theta+\operatorname{ctg}\theta\cos\varphi\,\partial/\partial\varphi); \]

\[ l_y=-i(\cos\varphi\,\partial/\partial\theta-\operatorname{ctg}\theta\sin\varphi\,\partial/\partial\varphi);\qquad l_z=-i\,\partial/\partial\varphi, \tag{2} \]

where \(r,\theta,\varphi\) are the spherical coordinates of the point \(x\).

The spin operator \(\bar s=\bar\sigma/2\) has the following projections on the coordinate axes:

\[ s_x=\frac12 \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}, \qquad s_y=\frac12 \begin{pmatrix} 0 & -i\\ i & 0 \end{pmatrix}, \qquad s_z=\frac12 \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}, \tag{3} \]

the term \(T(r)(\bar l,\bar s)\) takes into account the spin-orbit interaction of the scattered particles.

Introduce on the set of spinors

\[ u= \begin{pmatrix} u_1\\ u_2 \end{pmatrix} \]

a Hilbert space \(H\) with finite norm

\[ (u,u)=\|u\|^2=\int [\,|u_1|^2+|u_2|^2\,]\,dx, \tag{4} \]

where an integral without indicated limits of integration is taken over the whole three-dimensional space. The aim of the paper is a complete description of the spectrum of equation (1), the construction of eigenfunctions of the continuous spectrum for this equation, the proof of an expansion theorem for an arbitrary function from \(H\) in eigenfunctions, and the existence of the \(S\)-matrix and wave operators for the scattering problem under consideration. An analogous analysis for the Schrödinger operator without taking spin-orbit interaction into account in some infinite domains was carried out in \((^3)\).

§ 1. We shall call a spinor \(u\) finite if outside some domain the functions \(u_1(x)\) and \(u_2(x)\) are equal to zero. Denote by \(D^0\) the set of finite twice differentiable spinors.

Lemma 1. The Schrödinger operator \(\mathcal{L}\), considered on the set \(D^0\), is symmetric and semibounded. Its closure is a self-adjoint operator.

\[ \text{* Everything said below remains valid if } V(x) \text{ is a Hermitian matrix; in this case} \]

\[ |V(x)|\equiv \sum_{i,j=1}^{2}|V_{ij}(x)|. \]

Theorem 1. If \(V(x)\) and \(T(r)\) are real-valued functions, continuous outside and square-integrable inside some domain, tending to zero as \(|x|\to\infty\),

\[ \lim_{|x|\to\infty}\bigl[|V(x)|+|T(r)|\bigr]=0, \tag{5} \]

then the negative spectrum of the self-adjoint operator \(\mathcal L\) is discrete, and the positive spectrum is continuous.

If, moreover, the condition

\[ \lim_{r\to\infty}\sup_{\theta,\varphi} r\int_r^\infty \bigl[|V(t,\theta,\varphi)|+|T(t)|^2\bigr]\,dt=0, \tag{6} \]

is satisfied, then the negative spectrum of the operator \(\mathcal L\) consists of a finite number of eigenvalues.

Theorem 2. If

\[ \lim_{r\to\infty}\left[\sup_{|x|=r} r\,|V(x)|\right]<k;\qquad T(r)=0\quad \text{for } r>R_0, \tag{7} \]

then the operator \(\mathcal L\) has no positive eigenvalues. In other words, under conditions (7), every square-integrable solution of equation (1) is identically zero.

Let \(G(x,y,k)\) be the Green matrix for equation (1). Since the operator \(T(r)(\bar l\bar s)\) is a matrix differential operator, the Green function of equation (1) is a matrix

\[ G(x,y,k)= \begin{pmatrix} G_{11}(x,y,k) & G_{12}(x,y,k)\\ G_{21}(x,y,k) & G_{22}(x,y,k) \end{pmatrix}. \tag{8} \]

It satisfies the matrix equation:

\[ \mathcal LG=\delta(x-y)I, \tag{9} \]

where \(\delta(x-y)\) is the delta function, and \(I\) is the identity matrix

\[ I= \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix}. \tag{10} \]

The function \(G(x,y,k)\) admits analytic continuation into the half-plane \(\operatorname{Im} k>0\), i.e., all the functions \(G_{ij}(x,y,k)\), \(i,j=1,2\), have this property.

In what follows it is assumed that conditions (7) are satisfied and

\[ |V(x)|<\frac{c}{1+|x|^{3+a}},\qquad a>0. \tag{11} \]

Theorem 3. The function \(G(x,y,k)\) is a continuous function of the spectral parameter \(k\) in the half-plane \(\operatorname{Im} k\ge 0\)*.

It follows from Theorem 3 that

Corollary 1. The positive spectrum of the operator \(\mathcal L\) is absolutely continuous. The formula holds

\[ \frac{1}{\pi}\operatorname{Im}G(x,y,\sqrt{\lambda})=d\theta(x,y,\lambda)/d\lambda,\qquad \lambda\ge0. \tag{12} \]

Theorem 4. Let \(|y|\to\infty\), with the point \(y\) receding to infinity along a ray whose direction is characterized by the unit vector \(\bar\omega=\arg y\). Then

\[ G(x,y,k)\underset{|y|\to\infty,\arg y=\bar\omega}{=} \frac{e^{iky}}{4\pi |y|}\,\Phi(x,\bar\omega,k)(1+o(1)). \tag{13} \]

* If the operator \(\mathcal L\) has no negative discrete spectrum. In the contrary case, there exists a finite number of isolated simple poles lying on the positive part of the imaginary axis.

The matrix \(\Phi(x,\bar\omega,k)\) in our problem is analogous to the plane wave \(e^{-ik(\omega,x)}\) in the simplest scalar problem \((T=V=0)\). Note that

\[ \frac{e^{ik|x-y|}}{|x-y|} = \frac{e^{ik|y|}}{|y|}\,e^{-ik(\bar\omega x)}(1+o(1)). \]

\[ |y|\to\infty,\ \arg y=\bar\omega \]

Theorem 5 (on expansion in eigenfunctions). Let \(u\subset H\). The dual formulas hold

\[ u(x)=\frac{1}{(2\pi)^{3/2}}\int \Phi(x,\bar\omega\bar k)\,\hat u(\bar k)\,d\bar k+\sum_{p=1}^{n}c_p\varphi_p(x), \]

\[ \hat u(\bar k)=\frac{1}{(2\pi)^{3/2}}\int \Phi^*(x,\bar\omega,k)u(x)\,dx,\qquad c_p=(u,\varphi_p),\qquad \bar k=k\bar\omega, \tag{14} \]

where \(\varphi_p(x)\) are eigenfunctions of the negative spectrum of the operator \(\mathcal L\).

The columns of the matrix \(\Phi(x,\bar\omega,k)\) are eigenfunctions of the continuous spectrum of the operator \(\mathcal L\).

p. 2. For the proof of Theorems 2–5, some results are needed which, it seems to us, are of independent interest. Consider the operator equation in an abstract Hilbert space \(H\):

\[ d^2v/dr^2-Av/r^2+(k^2-Q)v=0,\qquad k^2>0. \tag{15} \]

In this equation \(r\) is a parameter, \(r\geqslant 0\), \(A\) is a self-adjoint, unbounded, positive operator,

\[ (Av,v)>0\quad \text{for } \|v\|>0. \tag{16} \]

The operator \(Q\) has the form

\[ Q=V(r)+T(r)R, \tag{17} \]

where \(V(r)\) and \(T(r)\) are bounded, continuous operators, and

\[ \limsup_{r\to\infty} r\,\|V(r)\|<2k\mu,\qquad \mu<\tfrac12;\qquad \|T(r)\|=0\quad \text{for } r>R_0=\mathrm{const}. \tag{18} \]

The operator \(R\) is self-adjoint, possibly unbounded, for \(r\leqslant R_0\).

Theorem 6. Under the assumptions made, every solution of equation (15) that is not identically zero satisfies the relations

\[ \lim_{R\to\infty}\int_R^{R+b} r^{2\mu+\varepsilon}\|v\|^2\,dr>\mathrm{const}>0, \tag{19} \]

\[ \liminf_{r\to\infty} r^{2\mu+\varepsilon}\bigl[\|v'\|^2+k^2\|v\|^2\bigr]>0 \tag{20} \]

for any \(\varepsilon>0\), any fixed \(b>0\).

If it is assumed that

\[ \int_{R_0}^{\infty}\|V(r)\|\,dr<\infty, \tag{21} \]

then in formulas (19), (20) one may put \(\varepsilon=\mu=0\).

Remark 1. Assertions (19)–(20) are uniqueness theorems, important for a rigorous mathematical analysis of the spectrum of the Schrödinger operator. With their help Theorem 2 is proved. The results of p. 2 are obtained on the basis of a generalization of the methods of papers \(({}^3,{}^4)\).

p. 3. Let the assumptions of p. 1 be fulfilled and, in addition, let the functions \(V(x)\) and \(T(r)\) be finite, i.e. vanish outside some region.

Theorem 7. Under the assumptions made, the Green matrix-function \(G(x,y,k)\) admits an analytic continuation to the entire plane of the complex variable \(k\). This continuation is analytic for \(\operatorname{Im} k \geqslant 0\)* and meromorphic for \(\operatorname{Im} k < 0\).

§ 4. Let the assumptions of § 1 be satisfied.

Theorem 8. In the scattering problem for the potential \(V(x)+T(r)(\bar l,\bar s)\), containing spin-orbit interaction, there exist wave operators \(W_\pm\) and a unitary scattering operator \(S\).

For the scalar problem in certain domains with an infinite boundary, an analogous theorem was proved in \((3\text{и},\ \text{к})\)**.

§ 5. If the operator \(\mathcal L\) has no discrete negative spectrum, then the limiting-amplitude principle is valid for this operator.

Theorem 9. Under the conditions of Theorem 7, the solution of the nonstationary problem

\[ u_{tt}+\mathcal L u=f(x)e^{i\omega t},\qquad |f(x)|<\frac{c}{1+|x|^{3+a}},\quad a>0, \tag{22} \]

\[ u\big|_{t=0}=u_t\big|_{t=0}=0 \tag{23} \]

admits the estimate, as \(t\to\infty\),

\[ u(x,t)=e^{i\omega t}v(x,\omega)+o(1), \tag{24} \]

where

\[ \mathcal L v-\omega^2 v=f(x). \tag{25} \]

Leningrad Institute
of Precision Mechanics and Optics

Received
15 IX 1965

REFERENCES

¹ A. S. Davydov, Theory of the Atomic Nucleus, Moscow, 1958. ² G. Ya. Lyubarskii, Group Theory and Its Applications in Physics, Moscow, 1958. ³ A. G. Ramm, a) DAN, 152, No. 2, 282 (1963); b) Vestn. LGU, ser. matem., mekh. i astr., 13, 3, 153 (1964); correction in 1, 1, 176 (1966); c) Vestn. LGU, ser. matem., mekh. i astr., I, II, 7, 2, 45 (1963); 19, 4, 69 (1963); d) Matem. sborn., 66 (108), 3, 321 (1965); e) Dokl. AN AzerbSSR, 21, 1, 3 (1965); UMN, 19, 5, 192 (1964); f) Izv. vyssh. uchebn. zaved., ser. matem., 2, 163 (1965); g) Abstracts of Reports at the Third All-Union Symposium on Wave Diffraction, “Nauka,” 1964; i) DAN, 163, No. 3, 584 (1965); k) Ukr. matem. zhurn., 17, 3, 192 (1965); l) DAN, 166, No. 6, 1319 (1966). ⁴ T. Kato, Comm. on Pure and Appl. Math., 12, 3, 403 (1959).

* See the footnote to Theorem 3.
** In paper \((3\text{к})\) there are a number of removable errors. In particular, in formula \((2')\), \(\Delta\) should stand instead of \(\nabla\). In paper \((3\text{и})\), in formula (8), \(D_x, D_y\) is printed; it should read \(D_x^2, D_y^2\).