Limiting-amplitude principle

A method for uniquely reconstructing solutions of stationary equations by a limit transition, as $t\to\infty$, of the amplitude of the solution of the corresponding non-stationary equation with zero initial data and a right-hand side of the form $f(x)e^{\pm i\omega t}$, periodic in $t$. If the limiting-amplitude principle holds, then the solution $v(x,t)$ of the non-stationary problem described has, as $t\to\infty$, the form

$$v(x,t)=u_\pm(x)e^{\pm i\omega t}+o(1),\label{*}\tag{*}$$

where $u_\pm$ is the solution to the stationary equation, which describes stable oscillations.

This principle was proposed at first [1] for the Helmholtz equation in $\mathbf R^n$,

$$(\Delta+k^2)u=f,$$

and it determines the same solution of this equation as the radiation conditions and the limit-absorption principle. Fulfillment of the limiting-amplitude principle has been investigated: for second-order equations with variable coefficients in the exterior of a bounded region (cf. [2], [3]); for the Helmholtz equation in certain regions with non-compact boundary (cf. [3], [4]); for the Cauchy–Poisson problem in a strip (cf. [5]); for certain higher-order equations (cf. [3], [6]); and for mixed problems in the exterior of a bounded region for equations and systems of equations of arbitrary order and with variable coefficients (cf. [7]). In the latter case the radiation and limit-absorption principles determine $2^\kappa$, $1<\kappa<\infty$, solutions to the stationary equation, while the limiting-amplitude principle determines only 2 of them. A statement of the limiting-amplitude principle that allows one to determine all $2^\kappa$ solutions has been given [8].

For the limiting-amplitude principle to hold it is necessary that $f$ is orthogonal to all eigen functions of the stationary problem. Therefore the principle does not hold in bounded regions. Let $P_\lambda$ be the operator corresponding to the stationary problem, depending polynomially on the spectral parameter $\lambda$, obtained from the mixed problem for a non-stationary equation by replacing in the equation and boundary conditions the differentiation operator $i\partial/\partial x$ by $\lambda$. The fulfillment of the limiting-amplitude principle for $P_\lambda$, $\lambda=\text{const}$, is related to the possibility of analytic continuation of the kernel of the resolvent $R_\lambda\equiv P_\lambda^{-1}$ onto the continuous spectrum and to the smoothness (in $\lambda$) of this continuation (cf. [3], [7]). If the kernel $R_\lambda$ allows analytic continuation across the continuous spectrum and if one has appropriate estimates, as $\lambda\to\infty$, then one can describe the asymptotics of the remainder $o(1)$, as $t\to\infty$, in \eqref{*}, and one can obtain asymptotic expansions, as $t\to\infty$, of solutions of other non-stationary problems (cf. [2], [7]). The properties of $R_\lambda$ mentioned above have been obtained in [7] for mixed problems in the exterior of a bounded region for equations and systems of equations of arbitrary order.

References

[1]	A.N. Tikhonov, A.A. Samarskii, "On the radiation principle" Zh. Eksper. i Teoret. Fiz. , 18 : 2 (1948) pp. 243–248 (In Russian)
[2]	O.A. Ladyzhenskaya, "On the limiting-amplitude principle" Uspekhi Mat. Nauk , 12 : 3 (1957) pp. 161–164 (In Russian)
[3]	D.M. Eidus, "The principle of limiting amplitude" Russian Math. Surveys , 24 : 3 (1969) pp. 97–167 Uspekhi Mat. Nauk , 24 : 3 (1969) pp. 91–156
[4]	A.G. Sveshnikov, "On the radiation principle" Dokl. Akad. Nauk SSSR , 73 : 5 (1950) pp. 917–920 (In Russian)
[5]	E.K. Isakova, "The limiting amplitude principle for the Cauchy–Poisson problem in a plane I" Differential Eq. , 6 : 1 (1970) pp. 45–55 Differentsial. Uravn. , 6 : 1 (1970) pp. 56–71
[6]	V.P. Mikhailov, "Stabilizing the solution of a certain nonsteadystate boundary value problem" Proc. Steklov Inst. Math. , 91 (1969) pp. 103–116 Trudy Mat. Inst. Steklov. , 91 (1967) pp. 100–112
[7]	B.R. Vainberg, "On the short wave asymptotic behaviour of solutions of stationary problems and the asymptotic behaviour as $t\to\infty$ of solutions of non-stationary problems" Russian Math. Surveys , 30 : 2 (1975) pp. 1–58 Uspekhi Mat. Nauk , 30 : 2 (1975) pp. 3–55
[8]	B.R. Vainberg, "The limiting amplitude principle" Izv. Vyzov. Mat. , 2 (1974) pp. 12–23 (In Russian)