M. M. Gekhtman

On the Principle of Limiting Amplitude

(Presented by Academician L. S. Pontryagin, 4 VI 1963)

Mathematics

Consider the Cauchy problem

\[ \frac{\partial^2 v(x,t)}{\partial t^2}+Lv=-f(x)e^{i\omega t}; \tag{1} \]

\[ v(x,0)=\frac{\partial v(x,0)}{\partial t}=0. \tag{2} \]

Here \(\omega>0\); \(f(x)\) is a finite sufficiently smooth function; \(L\) is a homogeneous positive definite elliptic operator with constant coefficients of order \(2m\); \(x=(x_1,\ldots,x_n)\) is a point of the \(n\)-dimensional space \(R_n\).

The Cauchy problem (1)—(2) is posed correctly; therefore there exists its unique solution \(v(x,t)\).

If there exists, uniformly in \(x\) varying in each bounded domain of \(R_n\), the limit \(u(x)\) equal to

\[ u(x)=\lim_{t\to\infty}v(x,t)e^{-i\omega t}, \tag{3} \]

then this limit is called the limiting amplitude \((^1)\).

The purpose of the present note is to study the behavior of the solution \(v(x,t)\) of the Cauchy problem (1)—(2) when \(t\) is sufficiently large and \(x\) varies in a bounded domain. In particular, the question of the existence of the limiting amplitude is considered.

In order to simplify the calculations, we shall assume that \(n\) is an odd number and \(L\equiv(-1)^m\Delta^m\), where \(\Delta\) is the Laplace operator. The problem is solved by the method of analytic continuation of the resolvent kernel \((^2)\).

We seek the solution of problem (1)—(2) by means of the Laplace transform with respect to \(t\), putting

\[ w(x,\lambda)=\int_0^\infty v(x,t)e^{-\lambda t}\,dt \qquad (\operatorname{Re}\lambda>0). \]

Then for \(w(x,\lambda)\) we obtain the equation

\[ Lw=-\lambda^2 w+\frac{f(x)}{\lambda-i\omega}. \tag{4} \]

In \(L^2(R_n)\) equation (4) has a unique solution \(w\), and for \(\operatorname{Re}\lambda>0\)

\[ w(x,\lambda)=\int_{R_n} H(x,y,\lambda)\frac{f(y)}{\lambda-i\omega}\,dy; \tag{5} \]

\(H(x,y,\lambda)\) is the resolvent kernel of the operator \(L\). We shall show that \(H(x,y,\lambda)\), as a function of \(\lambda\), admits analytic continuation into the domain \(-a\leq \operatorname{Re}\lambda\leq 0\) \((a>0)\), except, possibly, for the point \(\lambda=0\), and we assume that a cut has been made along the negative real semi-axis.

We shall seek \(H(x,\lambda)\) as a decreasing solution of the equation

\[ (L+\lambda^2)P(D)H_0(x,\lambda)=\delta(x); \tag{6} \]

here \(P(D)=[(-1)^p\Delta^p+1]\); \(\delta(x)\) is the Dirac function; \(H(x,\lambda)=P(D)H_0(x,\lambda^2)\), and the number \(p\) is chosen from the condition \(n-1<2m+2p\).

Expanding \(\delta(x)\) into plane waves \((^3)\) and using the Fourier transform, we obtain

\[ H(x,\lambda)=P(D)\int_{\Omega}\left[\int_{-\infty}^{\infty} \frac{s^{\,n-1}e^{isz}\,ds}{(s^{2m}+\lambda^2)(s^{2p}+1)} \right]d\omega . \tag{7} \]

In formula (7), \(\Omega\) is the unit sphere, \(\omega=(\omega_1,\ldots,\omega_n)\) is a point on this sphere, and

\[ z=\sum_{i=1}^{n}x_i\omega_i . \]

By virtue of formula (5),

\[ w(x,\lambda)=\int_{R_n}H_0(x,y,\lambda)\frac{P(D)f(y)}{\lambda-i\omega}\,dy . \tag{8} \]

Here we have used the fact that \(f(y)\) is a sufficiently smooth finite function, and \(H_0(x,y,\lambda)\) depends on the difference \(x-y\).

Applying the residue theorem, one can explicitly write the expression

\[ H_0(x,\lambda)=\int_{\Omega}\left[ \frac{\pi i}{m}\sum_{j=0}^{m-1} \frac{-\varepsilon_j^{\,n}\lambda^{(n-2m)/m}e^{\,i|z|\varepsilon_j\lambda^{1/m}}} {\lambda^{2p/m}\varepsilon_j^{2p}+1} + \frac{\pi i}{p}\sum_{k=0}^{p-1} \frac{-\delta_k^{\,n}e^{\,i|z|\delta_k}} {\lambda^2+\delta_k^{2m}} \right]d\omega . \tag{9} \]

In this formula \(\varepsilon_j=\sqrt[2m]{-1}\), \(\operatorname{Im}\varepsilon_j>0\), \(j=0,1,\ldots,m-1\); analogously, \(\delta_k=\sqrt[2p]{-1}\) are defined.

From formulas (8) and (9) it follows that \(w(x,\lambda)\) admits an analytic continuation into the region \(\operatorname{Re}\lambda\le 0\), except, perhaps, for the point \(\lambda=0\), and at the point \(\lambda=i\omega\), \(w(x,\lambda)\) has a simple pole with residue equal to 1. Along the negative real half-axis in the \(\lambda\)-plane a cut is made.

If \(a>0\) is fixed, then \(p\) can be chosen so large that

\[ |w(x,\lambda)|\le \frac{c(x)}{|\lambda|^2} \quad \text{for large } |\lambda|. \tag{10} \]

Theorem 1. Let \(n\) be an odd number, and let \(L\) be a homogeneous elliptic operator with constant coefficients of order \(2m\). In order that the limiting amplitude exist, it is sufficient that the kernel of the resolvent of the operator \(L\), \(H(x,\lambda)\), in a neighborhood of zero (with respect to the variable \(\lambda\)) satisfy the estimate

\[ |H(x,\lambda)|\le \frac{c(x)}{|\lambda|^{1-\varepsilon}} \quad (\varepsilon>0). \tag{11} \]

The proof is carried out by the method of contour integration in the same way as in \((^2)\), except that the contour of integration must not contain the point \(\lambda=0\) inside it.

From this theorem we obtain consequences.

Let \(n>2m\). Then the kernel of the resolvent admits an estimate of the form

\[ |H(x,\lambda)|\le c(x), \quad \text{if } |\lambda| \text{ is small}. \]

Consequently, the limiting amplitude exists. This is the case considered in \((^4)\).

Let \(m<n<2m\). In this case, when computing the resolvent kernel one may set \(p=0\). We obtain

\[ H(x,\lambda)=c_0\lambda^{(n-2m)/m}\sum_{j=0}^{m-1}-\varepsilon_j^{\,n}\int_0^1(1-p^2)^{(n-3)/2}e^{i\varepsilon_j\lambda^{1/m}|x|p}\,dp, \]

\[ |H(x,\lambda)|\leq \frac{c(x)}{|\lambda|^{2-n/m}}; \]

since \(n>m\), condition (11) is satisfied; the limiting amplitude exists, and we have obtained a strengthening of the results of [5].

Let \(n=m\). In this case

\[ v(x,t)=e^{i\omega t}\int_{\mathring R_n} H(x,y,i\omega)f(y)\,dy +c_0\int_{\mathring R_n} f(y)\,dy+o(1); \]

\(o(1)\) is a quantity tending to zero as \(t\to\infty\), \(c_0\ne0\). In order to preserve the limiting-amplitude principle, it is necessary additionally to require that

\[ \int_{\mathring R_n} f(y)\,dy=0. \]

If \(m>n\), then one may also take \(p=0\). \(H(x,\lambda)\), as a function of \(\lambda\), has at zero a singularity of type \(1/\lambda^\gamma\), where \(1<\gamma<2\). Expanding \(e^{i\varepsilon_j\lambda^{1/m}|x|}\) in a series, we obtain a finite number of terms whose singularity in \(\lambda\) at zero will be of the form \(1/|\lambda|^{1+\varepsilon}\) \((\varepsilon\geq0)\), while the remaining terms lead to the already considered singularity of the form \(1/|\lambda|^{1-\varepsilon_1}\) \((\varepsilon_1>0)\). Only the terms of the first type will give an asymptotic behavior growing as \(t\to\infty\).

Let us consider, for example, the asymptotic behavior corresponding to the first term of the expansion. Here the growth in \(t\) will be maximal. Clearly, this term has the form \(H_1(x,\lambda)=c/\lambda^\gamma\).

Carrying out the calculations, we obtain:

\[ v(x,t)=e^{i\omega t}\int_{\mathring R_n}H(x,y,i\omega)f(y)\,dy +c_0t^{1+\gamma}\int_{\mathring R_n}f(y)\,dy+\cdots+o(1). \]

Here terms are omitted which, as functions of \(t\), grow more slowly than \(t^{1+\gamma}\). Thus it has been proved:

Theorem 2. Let \(n\) be an odd number. In order that a limiting amplitude exist for a homogeneous positive definite elliptic operator \(L\), it is necessary and sufficient that \(n>m\).

I take this opportunity to express my gratitude to my scientific adviser A. G. Kostyuchenko for posing the problem, and to R. S. Ismagilov, who drew my attention to the importance of considering the resolvent kernel in a neighborhood of zero (with respect to the variable \(\lambda\)).

Moscow State University
named after M. V. Lomonosov

Received
16 V 1963

References

1. A. N. Tikhonov, A. A. Samarskii, Equations of Mathematical Physics, Moscow–Leningrad, 1953.
2. O. A. Ladyzhenskaya, Uspekhi Mat. Nauk, 12, no. 3 (1957).
3. I. M. Gelfand, G. E. Shilov, Generalized Functions and Operations on Them, issue 1, Moscow, 1959.
4. G. I. Bass, A. G. Kostyuchenko, Vestn. Moscow Univ., no. 5 (1959).
5. V. D. Bazdyrev, Izv. Vyssh. Uchebn. Zaved., Mathematics, no. 1, 8 (1963).
6. F. John, Plane Waves and Spherical Means, IL, 1958.