# 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  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. , ); for the Helmholtz equation in certain regions with non-compact boundary (cf. , ); for the Cauchy–Poisson problem in a strip (cf. ); for certain higher-order equations (cf. , ); 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. ). 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 .

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. , ). 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. , ). The properties of $R_\lambda$ mentioned above have been obtained in  for mixed problems in the exterior of a bounded region for equations and systems of equations of arbitrary order.

How to Cite This Entry:
Limiting-amplitude principle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Limiting-amplitude_principle&oldid=44712
This article was adapted from an original article by B.R. Vainberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article