Limit-absorption principle
A manner for uniquely finding solutions to equations analogous to the Helmholtz equation by introducing an infinitesimal absorption. Mathematically the principle is as follows. Let be an unbounded region in
, let
be the self-adjoint operator on
given by the differential expression
,
, and homogeneous boundary conditions on
and let
be a point in the continuous spectrum of
. Then for
the equation
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is uniquely solvable in , and in certain cases it is possible to find solutions
of the equation
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by the limit transition
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It is assumed here that has compact support and the convergence
, as
, is understood in the sense of
, where
is an arbitrary bounded set in
. Since
is a point of the continuous spectrum of
, the limit in
does not exist, in general.
The first limit-absorption principle was formulated for the Helmholtz equation in (cf. [1]):
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The solutions found using this principle are diverging or converging waves and satisfy the radiation conditions at infinity. These results were carried over (cf. [2], [3]) to elliptic boundary value problems in the exterior of bounded regions in
for an operator
![]() | (*) |
where the coefficients tend to constants sufficiently rapidly as
. In order that the limit-absorption principle holds in this case it is necessary that
is not an eigen value of
or that
is orthogonal to the eigen functions. A theorem of T. Kato (cf. [3]) gives sufficient conditions for the absence of eigen values in the continuous spectrum of the operator
. Such a theorem has been obtained for the operator (*) (cf. [3]). The limit-absorption principle has been substantiated for certain regions with non-compact boundary (cf. [3], [4]).
A limit-absorption principle and corresponding radiation conditions have been found for higher-order equations and for systems of equations (cf. [5]–[7]); they consist of the following. Let be an elliptic (or hypo-elliptic) operator satisfying: 1) the polynomial
has real coefficients; 2) the surface
,
, decomposes into connected smooth surfaces
,
, whose curvatures do not vanish; and 3)
on
. Suppose that an orientation is given on
, i.e. for each surface one has independently chosen a normal direction
. Let
, let
be a point on
at which
and
have identical direction and let
. Then the function
does satisfy the radiation conditions if it can be represented as
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These conditions determine a unique solution of the equation
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for any function with compact support. The limit-absorption principle for this equation is that this solution can be obtained as the limit, for
, of the unique solution
of the elliptic equation
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where has real coefficients and
on
. Depending on the choice of
,
, one obtains in the limit solutions satisfying the radiation conditions corresponding to some orientation of
. This principle has been substantiated for higher-order equations and systems with variable coefficients in the exterior of bounded regions (cf. [5]–[7]), as well as in the case of non-convex
. For such equations there is also a uniqueness theorem of Kato type.
References
[1] | W. [V.S. Ignatovskii] Ignatowsky, "Reflexion elektromagnetischer Wellen an einem Drahte" Ann. der Physik , 18 : 13 (1905) pp. 495–522 |
[2] | A.Ya. Povzner, "On the decomposition of arbitrary functions into eigenfunctions of the operator ![]() |
[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.V. Sveshnikov, "The limit absorption principle for a wave guide" Dokl. Akad. Nauk SSSR , 80 : 3 (1951) pp. 345–347 (In Russian) |
[5] | B.R. Vainberg, "Principles of radiation, limit absorption and limit amplitude in the general theory of partial differential equations" Russian Math. Surveys , 21 : 3 (1966) pp. 115–193 Uspekhi Mat. Nauk , 21 : 3 (1966) pp. 115–194 |
[6] | B.R. Vainberg, "On elliptic problems in unbounded domains" Math. USSR Sb. , 4 (1968) pp. 419–444 Mat. Sb. , 75 : 3 (1968) pp. 454–480 |
[7] | B.R. Vainberg, "Asymptotic methods in equations of mathematical physics" , Gordon & Breach (1988) (Translated from Russian) |
Limit-absorption principle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Limit-absorption_principle&oldid=47637