A. G. RAMM

ON NECESSARY AND SUFFICIENT CONDITIONS FOR THE VALIDITY OF THE LIMITING AMPLITUDE PRINCIPLE

(Presented by Academician I. G. Petrovskii, 15 I 1965)

1°. Let \(\mathscr L\) be a semibounded operator in a Hilbert space. Consider the nonstationary problems

\[ u_{tt}+\mathscr L u=f(x)e^{i\omega t}; \tag{1} \]

\[ u\big|_{t=0}=0,\qquad u_t\big|_{t=0}=0; \tag{2} \]

\[ u_{tt}+\mathscr L u=0; \tag{1^1} \]

\[ u\big|_{t=0}=0,\qquad u_t\big|_{t=0}=g(x). \tag{1^2} \]

Below we formulate conditions on the spectrum of the operator \(\mathscr L\) that are necessary and sufficient for the limiting amplitude principle to hold for the operator \(\mathscr L\) in the form \(\left({}^{1}\right)\)

\[ \frac1T \int_0^T u(x,t)e^{-i\omega t}\,dt-v(x,\omega)=o(1) \quad \text{as } T\to\infty . \tag{3} \]

We shall assume that the resolvent \((\mathscr L-\lambda I)^{-1}\) of the operator \(\mathscr L\) has the following properties:

A. The resolvent of the operator \(\mathscr L\) is an integral operator with kernel \(H(x,y,\lambda)\). The kernel is an analytic function of \(\lambda\) for \(\operatorname{Im}\lambda\ne0\). For \(\operatorname{Im}\lambda=0\) the function \(H(x,y,\lambda+i\varepsilon)\) \((\varepsilon>0,\ \lambda\ge0)\) tends uniformly to its limiting values \(H(x,y,\lambda)\), when \(x,y,\lambda\) vary in a bounded domain, \(|x-y|\ge\delta\). As \(|x-y|\to0\),
\[ |H(x,y,\lambda)|=O(1/|x-y|). \]

B. For \(\operatorname{Re}\lambda<0,\ \operatorname{Im}\lambda=0\), the function \(H(x,y,\lambda)\) has a finite number of poles.

Assumption B is sometimes replaced by the assumption

C. For \(\operatorname{Re}\lambda<0\), the function \(H(x,y,\lambda)\) is analytic in \(\lambda\).

If C is satisfied, then the operator \(\mathscr L\) has no spectrum to the left of zero; consequently,
\[ (\mathscr L u,u)>0 \]
(the equality sign is excluded by condition A).

D.
\[ \left|\int_D H(x,y,-p^2)f(y)\,dy\right| < \frac{c}{1+|p|^\gamma}, \qquad \operatorname{Re}p\ge0,\quad \gamma>0,\quad \operatorname{Im}p\to\infty \]

for smooth \(f(y)\) decreasing sufficiently rapidly, so that the integral written converges absolutely and uniformly with respect to \(p^2\), varying in a bounded domain \(\operatorname{Re}p\ge0\).

F.
\[ \left|\int_D [H(x,y,\lambda+h)-H(x,y,\lambda)]f(y)\,dy\right| \xrightarrow[|h|\to0]{}0, \qquad \operatorname{Re}\lambda\ge0 . \]

Assumption A is sometimes replaced by the assumption

A\(_1\). If the positive spectrum of the operator \(\mathscr L\) is continuous, then the resolvent kernel has continuous limiting values as \(\varepsilon\to0\), \(\operatorname{Im}\lambda=0,\ \operatorname{Re}\lambda\ge0\).

The content of the present note consists of the following theorems.

Theorem 1. Suppose that assumptions A, C, D, F are satisfied. Then, for the solution of problem (1)—(2), estimate (3) is valid. The function \(v(x,\omega)\) solves the stationary problem

\[ \mathcal{L}v-\omega^{2}v=f(x), \tag{4} \]

\(v(x,\omega)\) admits an analytic continuation into the half-plane \(\operatorname{Im}\omega>0\).

Theorem 2. Suppose that assumptions B, D are satisfied. The operator \(\mathcal{L}\) has no discrete negative spectrum if and only if the solution of problem \((1^1)—(1^2)\) admits the estimate

\[ \left|\int_{0}^{t} u(x,\tau)\,d\tau\right|=O(e^{\varepsilon t}),\qquad t\to\infty, \tag{5} \]

where \(\varepsilon>0\) is arbitrarily small.

Theorem 3. Suppose that assumptions C, D, A\(_1\) are satisfied. Then, for the operator \(\mathcal{L}\) to have no positive eigenvalues, it is necessary and sufficient that

\[ \left|\int_{0}^{t} u(x,\tau)e^{-i\lambda\tau}\,d\tau\right|=O(1),\qquad t\to\infty, \tag{6} \]

for all \(\lambda\geqslant 0\). In order that the point \(\lambda_0\geqslant 0\) not be a point of the discrete spectrum, it is sufficient that (6) hold for \(\lambda=\lambda_0\). (The function \(u(x,\tau)\) is the solution of problem \((1^1)—(1^2)\).)

Remark 1. In Theorems 2 and 3 it is assumed that, as \(g\), one may take any element from a set dense in the Hilbert space.

Theorem 4. Suppose that assumptions B, D, A\(_1\), F are satisfied. In order that the limiting amplitude principle in the form (3) hold for the operator \(\mathcal{L}\), it is necessary and sufficient that assumptions A and C hold.

Theorems analogous to Theorems 1 and 2 were obtained earlier by the author in \((^1)\) for the case \(\mathcal{L}=-\Delta+p(x)\).

\(2^0\). Let \(D\) be an infinite domain with smooth boundary \(\Gamma\) in three-dimensional space. Consider the integral operator

\[ Kf=\int_D K(x,y)f(y)\,dy. \tag{7} \]

Theorem 5. Suppose

\[ \iint_{D\,D}\left[|D_xK(x,y)|^2+|D_yK(x,y)|^2+p^2(x)|K(x,y)|^2\right]\,dx\,dy<\infty, \tag{8} \]

where

\[ p(x)>C(1+|x|)^{6+a},\qquad a>0. \tag{9} \]

Then \(K\) is a nuclear operator (the definition of a nuclear operator is given in \((^2)\)).

We use Theorem 5 to prove the following theorem:

Theorem 6. Let \(D\) be a plane domain with boundary \(\Gamma\), asymptotically approaching the boundary \(\Gamma_0\) of an angle \(D_0\), so that

\[ \rho(s,\Gamma)<\frac{C}{1+|s|^{7+\alpha}},\qquad \alpha>0, \tag{10} \]

where \(\rho(s,\Gamma)\) is the distance from the point \(s\subset\Gamma\) to \(\Gamma_0\).

Suppose

\[ \mathcal{L}u=-\Delta u+c(x)u, \tag{11} \]

where

\[ p^2(x)|c(x)|<\frac{c}{1+|x|^{2+b}},\qquad b>0. \tag{12} \]

Denote by \(\mathcal L\) the operator of the Dirichlet problem for the differential expression (11) in the domain \(D\), and by \(\mathcal L_0\) the operator of the Dirichlet problem in the domain \(D_0\) for the Laplace operator.

Under the assumptions made, there exist wave operators \(W_{\pm}(\mathcal L,\mathcal L_0)\) and the scattering matrix
\(S=W_+^*(\mathcal L,\mathcal L_0)W_-(\mathcal L,\mathcal L_0)\).

For definitions pertaining to the notion of wave operators, see \((^3)\). The resolvent kernel of the Schrödinger operator for the Dirichlet problem in the domain \(D\) was studied in \((^1)\) (see also \((^4,^5)\)).

Leningrad Institute
of Precision Mechanics and Optics

Received
7 I 1965

CITED LITERATURE

\(^1\) A. G. Ramm, DAN, 152, No. 2, 282 (1963).
\(^2\) I. M. Gelfand, N. Ya. Vilenkin, Some Applications of Harmonic Analysis. Rigged Hilbert Spaces, Moscow, 1962.
\(^3\) M. Sh. Birman, Izv. AN SSSR, ser. matem., 27, No. 4 (1963).
\(^4\) A. G. Ramm, Vestn. LGU, 7, No. 2, 45 (1963).
\(^5\) A. G. Ramm, Vestn. LGU, 19, No. 4, 69 (1963).