Full Text
Reports of the Academy of Sciences of the USSR
- Vol. 120, No. 6
MATHEMATICS
O. A. LADYZHENSKAYA and L. D. FADDEEV
ON THE THEORY OF PERTURBATIONS OF THE CONTINUOUS SPECTRUM
(Presented by Academician V. I. Smirnov on 17 II 1958)
K. O. Friedrichs \((^{1,2})\) investigated the character of the spectrum for the operator \(L = L_0 + \varepsilon K\), where \(L_0\) is the operator of multiplication by the independent variable in a Hilbert space of abstract functions, and \(K\) is an integral operator. Under the fulfillment of a number of conditions of smoothness and decay type, similar to conditions \((R)\) and \((K)\) (see below), and for sufficiently small \(\varepsilon\), he proved the unitary equivalence of the operators \(L\) and \(L_0\), and also investigated the behavior as \(|t| \to \infty\) of the solution of the Schrödinger equation with the operator \(L\). The main part of his investigation consists in the study of the integral equation
\[ r(\lambda,\mu)=k(\lambda,\mu)+i\pi\varepsilon k(\lambda,\mu)r(\mu,\mu)+\varepsilon P\int \frac{k(\lambda,\sigma)r(\sigma,\mu)}{\mu-\sigma}\,d\sigma \tag{I} \]
(the index \(P\) means that the integral is understood in the sense of the principal value) for the kernel \(r(\lambda,\mu)\) of the operator \(R\), by means of which the operator \(U\), effecting the unitary transformation of \(L\) into \(L_0\), is constructed. Friedrichs needed the smallness of \(\varepsilon\) in order to prove the solvability of equation (I). In the present paper the solvability of equation (I) is proved for arbitrary \(\varepsilon\) (we shall henceforth put \(\varepsilon=1\)).
- Basic concepts and notation. Let \(A\) be a complex Hilbert space of elements \(x,y,\ldots\) with scalar product \(xy\) and norm \(|x|=(xx)^{1/2}\). The set of measurable functions \(x(\lambda)\) of the real variable \(\lambda\), varying in the interval \(I\), with values in \(A\), for which * \(\int |x(\lambda)|^2\,d\lambda<\infty\), is a Hilbert space (which we shall denote by \(\mathcal H\)) if the scalar product
\[ (x(\lambda),y(\lambda))=\int x(\lambda)y(\lambda)\,d\lambda. \]
is introduced in it. On all functions for which
\[ \int \lambda^2 |x(\lambda)|^2\,d\lambda<\infty, \]
there is defined the operator \(L_0x(\lambda)=\lambda x(\lambda)\), which is self-adjoint in \(\mathcal H\). Let, further, \(k(\lambda,\mu)\), for fixed \(\lambda\) and \(\mu\) from \(I\), be a bounded operator in \(A\); by \(|k(\lambda,\mu)|\) we shall denote its norm, and by \(\overline{k(\lambda,\mu)}\) the adjoint operator. In what follows, the kernel \(k(\lambda,\mu)\) will be subject to the following conditions:
\[ \left|(1+|\lambda|^\beta+|\mu|^\gamma)k(\lambda,\mu)\right|\le c_1,\qquad 0<\beta,\gamma<1; \tag{R_1} \]
\[ \left|k(\lambda,\mu)-k(\lambda',\mu)\right|\le c_2|\lambda-\lambda'|^\beta; \tag{R_2} \]
\[ \left|(1+|\lambda|^\beta)\bigl(k(\lambda,\mu)-k(\lambda,\mu')\bigr)\right|\le c_3|\mu-\mu'|^\gamma; \tag{R_3} \]
\[ \left|k(\lambda,\mu)-k(\lambda',\mu)-k(\lambda,\mu')+k(\lambda',\mu')\right|\le c_4|\lambda-\lambda'|^\beta|\mu-\mu'|^\gamma; \tag{R_4} \]
\[ k(\lambda,\mu)=0,\quad \text{if } \mu \text{ is on the boundary of } I; \tag{R_5} \]
* All integrals in what follows are extended over the interval \(I\).
\(k(\lambda,\mu)\) is a completely continuous operator in \(A\) for all \(\lambda\) and \(\mu\) from \(I\),
\((T)\)
\[ k(\lambda,\mu)=\overline{k(\mu,\lambda)}. \tag{K} \]
We shall call the operator kernel \(r(\lambda,\mu)\) a kernel of class (R) if it satisfies conditions (R), except for the boundedness of \(|\mu^\gamma r(\lambda,\mu)|\).
Lemma 1. If conditions \((R_1)\) and \((R_2)\) are satisfied, with \(\beta>1/2\), then on every function from the domain of definition of \(L_0\) the operator
\[ Kx(\lambda)=\int k(\lambda,\mu)x(\mu)\,d\mu \]
is defined; and if condition (K) is also satisfied, then the operator \(L=L_0+K\) is self-adjoint in \(H\).
On the basis of our investigation of equation (I), carried out following Friedrichs’ method, we can prove the following assertion:
Theorem 1. Suppose that conditions (R), (T), and (K) are satisfied, with \(\beta>1/2\). Then the operator \(L\) has continuous spectrum on the interval \(I\) and at most a finite number of eigenvalues of finite multiplicity, which may lie both inside and outside the interval \(I\). The part of the operator \(L\) acting in the invariant subspace corresponding to the continuous spectrum is unitarily equivalent to the operator \(L_0\); i.e., there exists an operator \(U\) possessing the following properties:
\[ LU=UL_0;\qquad U^*U=1;\qquad UU^*=1-P. \tag{U} \]
Here \(P\) is the orthogonal projector onto the proper subspace of the operator \(L\), corresponding to the discrete spectrum.
In particular, the properties (U) are possessed by the operators
\[ U^{(\pm)}=\lim_{t\to\pm\infty} e^{-iLt}e^{iL_0t}, \]
where these limits exist in the strong sense. The operators \(U^{(\pm)}\) are related to one another by the formula
\[ U^{(+)}=U^{(-)}S, \]
where \(S\) is a unitary operator commuting with \(L_0\).
Let us note that the operator \(S\) has important significance in quantum mechanics—it is the so-called scattering operator, or \(S\)-matrix.
2. We now indicate the main stages of our investigation of equation (I). It is convenient to consider it in the space of functions \(u(\lambda)\) with values in \(A\), which satisfy a Hölder condition with some exponent \(\alpha\), \(0<\alpha<1\). The set of functions for which
\[ \|u\|_\alpha=\sup_\lambda |u(\lambda)|+\sup_{\lambda\lambda'}\frac{|u(\lambda)-u(\lambda')|}{|\lambda-\lambda'|^\alpha}+\sup_\lambda |\lambda^\alpha u(\lambda)|<\infty, \]
is a complete Banach space if \(\|u\|_\alpha\) is taken as the norm. We shall denote it by \(B_\alpha\). Consider in \(B_\alpha\) the operator
\[ T_\omega u(\lambda)=i\pi k(\lambda,\omega)u(\omega)+P\int \frac{k(\lambda,\sigma)u(\sigma)}{\omega-\sigma}\,d\sigma. \tag{1} \]
Lemma 2. Suppose that conditions (R) are satisfied. Then the operator \(T_\omega\) is defined on every function from \(B_\alpha\) and is bounded in the norm of \(B_\beta\). Moreover,
\[ \|T_\omega u\|_\beta \leqslant c_\omega \|u\|_\alpha, \]
where \(c_\omega \to 0\) as \(|\omega|\to\infty\), if \(J\) is bounded, and
\[ \|(T_\omega-T_{\omega'})u\|_\beta \le c|\omega-\omega'|^\delta \|u\|_\alpha,\qquad \delta=\min(\gamma,\alpha). \]
Lemma 3. Suppose the conditions of Lemma 2 are satisfied with \(\beta>\alpha\), and condition (T) is satisfied. Then the operator \(T_\omega\) is completely continuous in \(B_\alpha\).
We now consider the structure of the eigenfunctionals of the operator \(T_\omega^*\). Let \(T_\omega^* l_\omega=l_\omega\). This means that for every \(u\in B_\alpha\) we have:
\[ (l_\omega,u)=\left(l_\omega,i\pi k(\lambda,\omega)u(\omega)+P\int \frac{k(\lambda,\sigma)u(\sigma)}{\omega-\sigma}\,d\sigma\right). \]
If conditions \((R_1)\) and \((R_2)\) are satisfied, then for fixed \(\mu\) and arbitrary \(x\in A\), \(k(\lambda,\mu)x\) is an element of \(B_\beta\) and, all the more, of \(B_\alpha\). But then the expression \((l_\omega,k(\lambda,\mu)x)\) is meaningful, and it defines a linear functional in \(A\). By Riesz’ theorem,
\[ (l_\omega,k(\lambda,\omega)x)=\varphi_\omega(\mu)x. \tag{2} \]
This equality defines the function \(\varphi_\omega(\mu)\) with values in \(A\). If conditions \((R_1)\) and \((R_3)\) are satisfied, then \(\varphi_\omega\in B_\gamma\) and
\[ (l_\omega,u)=i\pi\varphi_\omega(\omega)u(\omega)+P\int \frac{\varphi_\omega(\sigma)u(\sigma)}{\omega-\sigma}\,d\sigma. \tag{3} \]
If here, as \(u(\lambda)\), we take \(k(\lambda,\mu)x\), then after simple transformations we obtain for \(\varphi_\omega(\mu)\) the equation
\[ \varphi_\omega(\mu)=-i\pi \overline{k(\omega,\mu)}\,\varphi_\omega(\omega)+P\int \frac{k(\sigma,\mu)\varphi_\omega(\sigma)}{\omega-\sigma}\,d\sigma. \]
If, in addition, condition (K) is satisfied, then the equation for \(\varphi_\omega(\mu)\) can be rewritten in the following form:
\[ \varphi_\omega(\mu)=T_\omega\varphi_\omega(\mu)-2\pi i k(\mu,\omega)\varphi_\omega(\mu). \]
From this it is not hard to see that \(\varphi_\omega(\omega)=0\). Indeed,
\[ (l_\omega,\varphi_\omega)=(l_\omega,T_\omega\varphi_\omega)-2\pi i(l_\omega,k(\mu,\omega)\varphi_\omega(\omega)) =(l_\omega,\varphi_\omega)-2\pi i|\varphi_\omega(\omega)|^2. \]
We obtain the following result:
Lemma 4. Suppose conditions (R), (T), and (K) are satisfied. Then the eigenfunctionals of the operator \(T_\omega^*\) corresponding to the eigenvalue \(1\) are constructed by formula (3) by means of the function \(\varphi_\omega(\mu)\in B_\gamma\), defined by equality (2), and moreover \(\varphi_\omega(\omega)=0\).
On the basis of Lemmas 2–4 the following theorem is proved:
Theorem 2. Under the fulfillment of conditions (R), (T), and (K), equation (I) is always solvable, and the solution \(r(\lambda,\mu)\) belongs to the class (R) and is a completely continuous operator in \(A\).
- As an example, consider the differential operator \(Mu=-\Delta u+q(x)u\) in the whole three-dimensional space \(E_3\). This operator is unitarily equivalent to an operator of type \(L\) of the general theory, where the interval \(I=(0,\infty)\), and the space \(A\) is the space of square-integrable functions on the unit sphere. The operator \(k(\lambda,\mu)\), for fixed \(\lambda\) and \(\mu\), is an integral operator on the unit sphere with kernel
\[ k(\lambda,\mu;\alpha,\beta)=\left(\frac{1}{2\pi}\right)^3 \frac{(\lambda\mu)^{1/4}}{2}\int_{E_3} q(x)e^{i(\sqrt{\lambda}\alpha-\sqrt{\mu}\beta,x)}\,dx; \]
here \(\alpha\) and \(\beta\) are unit vectors.
When a number of conditions of the type of smoothness and decrease of the function \(q(x)\) are satisfied for \(k(\lambda,\mu)\), the conditions (R) and (T) are satisfied and, if \(q(x)\) is real, then condition (K) is satisfied, so that the assertions of Theorems 1 and 2 hold. In particular, for the solution of the Schrödinger equation
\[ i\frac{\partial}{\partial t} z(\lambda,\alpha;t) = \lambda z(\lambda,\alpha;t) + \int_0^\infty d\mu \int d\beta\, k(\lambda,\mu;\alpha,\beta)\,z(\mu,\beta;t) \]
there exist the limits \(\lim_{t\to\pm\infty} e^{i\lambda t} z(\lambda,\alpha;t)=z_{\pm}(\lambda,\alpha)\) in the mean with respect to \(\lambda\) and \(\alpha\), if \(z(\lambda,\alpha;0)\) is orthogonal to the eigenfunctions of the discrete spectrum of the operator \(L\), and \(z_+(\lambda,\alpha)=S(\lambda)z_-(\lambda,\alpha)\). For fixed \(\lambda\), \(S(\lambda)\) is a unitary operator on the unit sphere. This is the so-called scattering operator of the operator \(M\).
Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
10 II 1958
References
¹ K. O. Friedrichs, Math. Ann., 115, No. 2, 249 (1938). ² K. O. Friedrichs, Comm. Pure and Appl. Math., 1, No. 4, 361 (1948).