D. M. Eidus

On the Principle of Limiting Amplitude

(Presented by Academician V. I. Smirnov on 27 IV 1964)

1. Let \(H\) be a Hilbert space; \(G\) a self-adjoint operator in \(H\), \(G \geq 0\); \(R_z\) the resolvent of the operator \(G\); \(H_0\) a subspace in \(H\); \(P\) the projection operator onto \(H_0\). Let \(w(t)\), where \(0 < t < \infty\), be a function whose values are elements of \(H\), having derivatives \(dw/dt\), \(d^2w/dt^2\) and satisfying the following equation and initial conditions:

\[ \frac{d^2 w}{dt^2} + Gw = f e^{-i\sqrt{\lambda}\,t}; \tag{1} \]

\[ \left. w \right|_{t=0} = 0, \qquad \left. \frac{dw}{dt} \right|_{t=0} = 0, \tag{2} \]

where \(\lambda < 0,\ f \in H\). The derivative \(dw/dt\) is assumed to be continuous for \(0 \leq t < \infty\), and the derivative \(d^2w/dt^2\) for \(0 < t < \infty\). Continuity and differentiability are understood in the strong sense. It is known that problem (1), (2) is uniquely solvable.

Theorem 1. Let the element \(f\) be such that the following conditions are satisfied:

1) For every \(\sigma > 0\) there exist the limits \(P R_z f \to v^+(\sigma)\) as \(z \to \sigma\), \(\operatorname{Im} z > 0\), and \(P R_z f \to v^-(\sigma)\) as \(z \to \sigma\), \(\operatorname{Im} z < 0\).

2) The function \(\theta(\sigma) = v^+(\sigma) - v^-(\sigma)\) satisfies the Hölder condition on every interval \(a \leq \sigma \leq b,\ a > 0\).

Then:

a) If the function \(\theta(\sigma)\) on some interval \(0 < \sigma \leq \delta\) satisfies the inequality

\[ \| \theta(\sigma) \| \leq c \sigma^{-\alpha}, \tag{3} \]

where \(0 \leq \alpha < 1/2\), then for every \(\lambda > 0\), as \(t \to +\infty\),

\[ e^{i\sqrt{\lambda}\,t} Pw \to v^+(\lambda). \]

b) If on the interval \(0 < \sigma \leq \delta\)

\[ \theta(\sigma) = \frac{\theta_0}{\sigma^\alpha} + \frac{\theta_1(\sigma)}{\sigma^\beta}, \tag{4} \]

where \(\beta < 1/2,\ \alpha = 1/2,\ \| \theta_1(\sigma) \| \leq C\), then for every \(\lambda > 0\), as \(t \to +\infty\),

\[ Pw - \left( e^{-i\sqrt{\lambda}\,t} v^+(\lambda) - \frac{\theta_0}{2\sqrt{\lambda}} \right) \to 0. \]

c) If on the interval \(0 < \sigma \leq \delta\) (4) holds, where \(1/2 < \alpha < 1,\ \beta < \alpha,\ \| \theta_1(\sigma) \| \leq C\), then for every \(\lambda > 0\)

\[ Pw = t^{2\alpha-1}(\varkappa \theta_0 + \varphi(t)), \]

where \(\varkappa = \varkappa_0/\sqrt{\lambda}\) (\(\varkappa_0\) is an absolute constant), \(\varphi(t) \to 0\) as \(t \to +\infty\).

Moreover, if inequality (3) holds for some \(\alpha < 1\), then for \(s = 1,2\) and \(t \to +\infty\) one has

\[ e^{i\sqrt{\lambda}\,t} P \frac{d^s w}{dt^s} \to \left(-i\sqrt{\lambda}\right)^s v^+(\lambda). \]

1. Consider in an infinite domain \(\Omega\) of \(n\)-dimensional space \(E_n\) the equation

\[ \frac{\partial^2 w}{\partial t^2}+gw=e^{-i\sqrt{\lambda}\,t}f(x) \tag{5} \]

with zero initial and homogeneous boundary conditions. Here \(\lambda>0\), \(g\) is an elliptic differential operator of order \(2m\). It is said that for equation (5) the limiting-amplitude principle holds if

\[ \lim_{t\to+\infty} e^{i\sqrt{\lambda}\,t}w=v, \tag{6} \]

where \(v\) is a solution of the equation

\[ gv=\lambda v+f. \tag{7} \]

A. N. Tikhonov and A. A. Samarskii, in the work (1), formulated this principle and proved that it holds in the case \(\Omega=E_3,\ g=-\Delta\). O. A. Ladyzhenskaya, in the article (2), proved that (6) holds in the case \(\Omega=E_3,\ g=-\Delta+q(x)\), \(q(x)\) is finite and the operator \(g\) has no points of the discrete spectrum. G. I. Bass and A. G. Kostyuchenko (3) gave a proof in the case when \(\Omega=E_n,\ 2m<n,\ g\) is a self-adjoint operator with constant coefficients.

In the present note the case of the operator

\[ g=(-1)^m \sum_{|k|=|l|=m} D^k\bigl(a_k^l(x)D^l\bigr)+q(x), \]

is considered, where

\[ k=(k_1,\ldots,k_n),\quad |k|=\sum_{s=1}^n k_s,\quad D^k=\frac{\partial^{|k|}}{\partial x_1^{k_1}\cdots \partial x_n^{k_n}}, \]

under the boundary conditions

\[ \left.\frac{\partial^p u}{\partial \nu^p}\right|_{\Gamma}=0, \tag{8} \]

where \(p=0,1,\ldots,m-1\); \(\Gamma\) is the boundary of the domain \(\Omega\); \(\nu\) is the normal to \(\Gamma\). The boundary \(\Gamma\) (if it exists) of the infinite domain \(\Omega\) will be assumed to consist of a finite number of closed nonintersecting \((n-1)\)-dimensional surfaces. Denote by \(S_\rho\) the sphere of radius \(\rho\) with center at the origin of coordinates, and by \(\Omega_\rho\) and \(B_\rho\) the sets of all points of \(\Omega\) lying respectively inside and outside \(S_\rho\). Let \(a_k^l=\overline{a_l^k}\), \(q=\overline q\); \(a_k^l=a_l^k\); \(a_k^l(x)\in C^m(\overline{\Omega})\); \(q(x)\) is measurable and bounded in \(\Omega\). Assume that for all real \(\xi_k=\xi_{k_1\ldots k_n}\) and for all \(x\in\Omega\) the inequality

\[ \sum_{|k|=|l|=m} a_k^l(x)\xi_k\xi_l \ge \gamma \sum_{|k|=m}\xi_k^2 \]

holds, where \(\gamma>0\) is a constant. Suppose that there exists \(\rho_0>0\) such that in \(B_{\rho_0}\) \(f(x)=0,\ q(x)=0\),

\[ a_k^l(x)=\frac{m!}{k_1!\cdots k_n!}\delta_{k_1}^{\,l_1}\cdots \delta_{k_n}^{\,l_n} \]

(and thus in \(B_{\rho_0}\), \(g=(-\Delta)^m\)). In addition, let \(f\in L_2(\Omega)\). Below we shall assume that \(\Gamma\) lies inside \(S_{\rho_0}\) and \(\rho>\rho_0\).

Let us introduce some classes of functions. Denote by \(\overset{0}{D}_{\rho}\) the closure, in the metric of \(W_2^{(m)}(\Omega_\rho)\), of the set of those functions each of which is defined in \(\Omega_\rho\) and is identically zero in some strip along \(\Gamma\). By \(\overset{0}{D}\) denote the class of those functions which are defined in \(\Omega\) and belong to \(\overset{0}{D}_{\rho}\) for every \(\rho>\rho_0\). Condition (8) will be understood in the sense \(u\in\overset{0}{D}\). Denote by \(\overline{W}_2^{(k)}\) the set of functions each of which is defined

in \(\Omega\) and in any finite interior subdomain \(\omega\) belongs to \(W_2^{(k)}(\omega)\).

Put \(\Phi = \overset{\circ}{D}\cap \overline{W}_2^{(2m)}\) and define the differential operator \(g\) on the set \(\Phi\). We shall say that a sequence \(\varphi_k\in \Phi\) converges in the sense of \(\Phi\) to a function \(\varphi\), if \(\varphi_k\to \varphi\) in any finite interior subdomain \(\omega\) in the metric \(W_2^{(2m)}(\omega)\) and, moreover, \(\varphi_k\to \varphi\) in any \(\Omega_\rho\) in the metric \(W_2^{(2)}(\Omega_\rho)\). Then \(\varphi\in \Phi\). Introduce the operator \(G\) on functions \(u\in \Phi\cap L_2(\Omega)\), \(gu\in L_2(\Omega)\), by means of the equality \(Gu=gu\). Then \(G\) is a self-adjoint operator in \(L_2(\Omega)\). Let \(R_z\) be the resolvent of the operator \(G\). Put \(u_z=R_z f\), where \(\operatorname{Im} z\ne 0\). Denote by \(A\) the set of all \(\lambda>0\) which are points of discontinuity of the spectral function of the operator \(G\). Examples show that, generally speaking, \(A\ne (0,\infty)\). However, in the case \(g=(-\Delta)^m\) one has \(A=(0,\infty)\).

Theorem 2. Let \(\lambda\in A\), \(\operatorname{Im} z>0\). Then, as \(z\to\lambda\), \(u_z\to u_\lambda^+\) in the sense of \(\Phi\), where \(u_\lambda^+\) is the solution of equation (7) satisfying at infinity the radiation condition of I. N. Vekua \((^4)\).

If \(\operatorname{Im} z<0\), then \(u_z\to u_\lambda^-\) as \(z\to\lambda\), where \(u_\lambda^-\) is the solution of equation (7) satisfying the conjugate radiation condition.

Theorem 3. Let \([a,b]\subset A\). Then for any \(\rho>\rho_0\) there exists a constant \(c_\rho\) such that for any \(\lambda_1,\lambda_2\) from \([a,b]\) the inequality holds
\[ \|u_{\lambda_2}^{\pm}-u_{\lambda_1}^{\pm}\|_{L_2(\Omega_\rho)} \leq c_\rho |\lambda_2-\lambda_1|. \]

Theorem 4. Let \(q(x)\geq 0\) in \(\Omega\), and suppose at least one of the following conditions is fulfilled: 1) \(2m<n\); 2) \(q(x)>0\) on a set of positive measure; 3) \(\Omega\ne E_n\).

Then there exists a \(\delta>0\) such that \((0,\delta)\subset A\). Moreover, as \(\lambda\to +0\), \(u_\lambda^+\to u_0\), \(u_\lambda^-\to u_0\) in the sense of \(\Phi\), where \(u_0\) is the solution of the equation
\[ gu=f. \]

Let us note that \(u_0\) satisfies the usual condition at infinity which is imposed for the polyharmonic equation.

Theorem 5. Let \(\Omega=E_n\), \(q(x)=0\) in \(E_n\), \(2m\geq n\), \((0,\delta)\subset A\). Then:

1) for \(2m=n\)
\[ u_\lambda^\pm=q^\pm \ln \frac{1}{\lambda}+\psi_\lambda^\pm; \]

2) for \(2m-n=1\)
\[ u_\lambda^\pm=q^\pm \lambda^{-1/2m}+\psi_\lambda^\pm; \]

3) for \(2m-n\geq 2\)
\[ u_\lambda^\pm=q^\pm \lambda^{n/2m-1}+\psi_\lambda^\pm \lambda^{(n+2)/2m-1}, \]
where
\[ \|\psi_\lambda^\pm\|_{L_2(\Omega_\rho)}\leq C(\rho) \]
for \(0<\lambda<\delta\); \(q^\pm=\tau^\pm\int_\Omega f\,dx\); \(\tau^+\), \(\tau^-\) are constants depending only on \(m,n\), and \(\tau^+\ne \tau^-\).

We now consider the solution \(w(t)\) of problem (5) with zero initial conditions and the homogeneous boundary conditions introduced above.

Theorem 6. Let \(A=(0,\infty)\), \(q(x)\geq 0\) in \(\Omega\), and suppose at least one of the following conditions is fulfilled: 1) \(m<n\); 2) \(q(x)>0\) on a set of positive measure; 3) \(\Omega\ne E_n\).

Then for any \(\lambda>0\), as \(t\to+\infty\),
\[ e^{i\sqrt{\lambda}\,t}w\to u_\lambda^+. \]

in the sense of \(\Phi\). Moreover, in any domain \(\Omega_\rho\), as \(t\to+\infty\),

\[ e^{i\sqrt{\lambda}t}\frac{d^s w}{dt^s}\to (i\sqrt{\lambda})^s u_\lambda^+ \tag{9} \]

in the sense of \(\mathscr L_2(\Omega_\rho)\), where \(s=1,2\).

Theorem 7. Let \(A=(0,\infty)\), \(\Omega=E_n\), \(q(x)=0\) in \(E_n\). Then:

1) If \(m=n\), then for any \(\lambda>0\)

\[ w(t)=e^{-i\sqrt{\lambda}t}u_\lambda^+ + \frac{\mu}{\sqrt{\lambda}}\int_\Omega f\,dx+\varphi(t). \]

2) If \(m>n\), then for any \(\lambda>0\)

\[ w(t)=t^{1-n/m}\left(\frac{\mu}{\sqrt{\lambda}}\int_\Omega f\,dx+\varphi(t)\right), \]

where \(\|\varphi(t)\|_{\mathscr L_2(\Omega_\rho)}\to 0\) as \(t\to+\infty\); \(\mu\) is a constant depending only on \(m,n\); \(\mu\ne0\). Moreover, for \(m\ge n\), (9) holds.

Theorems 6 and 7 are proved with the aid of Theorem 1, whose conditions turn out to be satisfied by virtue of Theorems 2, 3, 4, 5. Here the role of the space \(H\) is played by the space \(\mathscr L_2(\Omega)\), and the role of the subspace \(H_0\) by the set of functions from \(\mathscr L_2(\Omega)\) identically equal to zero in \(B_\rho\).

Received
23 IV 1964

REFERENCES

\(^{1}\) A. N. Tikhonov, A. A. Samarskii, ZhETF, 18, no. 2, 243 (1948).
\(^{2}\) O. A. Ladyzhenskaya, UMN, 12, no. 3, 164 (1957).
\(^{3}\) G. I. Bass, A. G. Kostyuchenko, Vestn. Mosk. Univ., ser. matem. i mekh., No. 5, 153 (1959).
\(^{4}\) I. N. Vekua, Tr. Matem. Inst. AN GruzSSR, No. 12, 104 (1943).