View source for Limiting-amplitude principle

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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 [[#References|[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|radiation conditions]] and the [[Limit-absorption principle|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. [[#References|[2]]], [[#References|[3]]]); for the Helmholtz equation in certain regions with non-compact boundary (cf. [[#References|[3]]], [[#References|[4]]]); for the Cauchy–Poisson problem in a strip (cf. [[#References|[5]]]); for certain higher-order equations (cf. [[#References|[3]]], [[#References|[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. [[#References|[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 [[#References|[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. [[#References|[3]]], [[#References|[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. [[#References|[2]]], [[#References|[7]]]). The properties of $R_\lambda$ mentioned above have been obtained in [[#References|[7]]] for mixed problems in the exterior of a bounded region for equations and systems of equations of arbitrary order.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Tikhonov,   A.A. Samarskii,   "On the radiation principle"  ''Zh. Eksper. i Teoret. Fiz.'' , '''18''' :  2  (1948)  pp. 243–248  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  O.A. Ladyzhenskaya,   "On the limiting-amplitude principle"  ''Uspekhi Mat. Nauk'' , '''12''' :  3  (1957)  pp. 161–164  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.G. Sveshnikov,   "On the radiation principle"  ''Dokl. Akad. Nauk SSSR'' , '''73''' :  5  (1950)  pp. 917–920  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  B.R. Vainberg,   "The limiting amplitude principle"  ''Izv. Vyzov. Mat.'' , '''2'''  (1974)  pp. 12–23  (In Russian)</TD></TR></table>