View source for Limiting-amplitude principle

A method for uniquely reconstructing solutions of stationary equations by a limit transition, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l0589301.png" />, of the amplitude of the solution of the corresponding non-stationary equation with zero initial data and a right-hand side of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l0589302.png" />, periodic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l0589303.png" />. If the limiting-amplitude principle holds, then the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l0589304.png" /> of the non-stationary problem described has, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l0589305.png" />, the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l0589306.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l0589307.png" /> is the solution to the stationary equation, which describes stable oscillations.

This principle was proposed at first [[#References|[1]]] for the Helmholtz equation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l0589308.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l0589309.png" /></td> </tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l05893010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l05893011.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l05893012.png" /> solutions has been given [[#References|[8]]].

For the limiting-amplitude principle to hold it is necessary that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l05893013.png" /> is orthogonal to all eigen functions of the stationary problem. Therefore the principle does not hold in bounded regions. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l05893014.png" /> be the operator corresponding to the stationary problem, depending polynomially on the spectral parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l05893015.png" />, obtained from the mixed problem for a non-stationary equation by replacing in the equation and boundary conditions the differentiation operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l05893016.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l05893017.png" />. The fulfillment of the limiting-amplitude principle for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l05893018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l05893019.png" />, is related to the possibility of analytic continuation of the kernel of the resolvent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l05893020.png" /> onto the continuous spectrum and to the smoothness (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l05893021.png" />) of this continuation (cf. [[#References|[3]]], [[#References|[7]]]). If the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l05893022.png" /> allows analytic continuation across the continuous spectrum and if one has appropriate estimates, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l05893023.png" />, then one can describe the asymptotics of the remainder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l05893024.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l05893025.png" />, in (*), and one can obtain asymptotic expansions, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l05893026.png" />, of solutions of other non-stationary problems (cf. [[#References|[2]]], [[#References|[7]]]). The properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l05893027.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058930/l05893028.png" /> 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>