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 $ \Omega $ be an unbounded region in $ \mathbf R ^ {n} $, let $ P $ be the self-adjoint operator on $ L _ {2} ( \Omega ) $ given by the differential expression $ P ( x, \partial / \partial x) $, $ x \in \Omega $, and homogeneous boundary conditions on $ \Omega $ and let $ \lambda $ be a point in the continuous spectrum of $ P $. Then for $ \epsilon \neq 0 $ the equation

$$ Pu _ \epsilon = \ ( \lambda + i \epsilon ) u _ \epsilon + f $$

is uniquely solvable in $ L _ {2} ( \Omega ) $, and in certain cases it is possible to find solutions $ u = u _ \pm $ of the equation

$$ Pu = \lambda u + f $$

by the limit transition

$$ u _ \pm = \ \lim\limits _ {\epsilon \rightarrow \pm 0 } \ u _ \epsilon . $$

It is assumed here that $ f $ has compact support and the convergence $ u _ \epsilon \rightarrow u _ \pm $, as $ \epsilon \rightarrow \pm 0 $, is understood in the sense of $ L _ {2} ( \Omega ^ \prime ) $, where $ \Omega ^ \prime $ is an arbitrary bounded set in $ \Omega $. Since $ \lambda $ is a point of the continuous spectrum of $ P $, the limit in $ L _ {2} ( \Omega ) $ does not exist, in general.

The first limit-absorption principle was formulated for the Helmholtz equation in $ \mathbf R ^ {2} $( cf. [1]):

$$ ( \Delta + k ^ {2} ) u = - f,\ \ \Omega = \mathbf R ^ {2} , $$

$$ P = - \Delta ,\ \lambda = - k ^ {2} < 0. $$

The solutions $ u _ \pm $ 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 $ \mathbf R ^ {n} $ for an operator

$$ \tag{* } P \left ( x,\ { \frac \partial {\partial x } } \right ) = - \sum _ {k, j = 1 } ^ { n } { \frac \partial {\partial x _ {k} } } \left ( a _ {kj} \frac \partial {\partial x _ {j} } \right ) + q ( x), $$

where the coefficients $ a _ {kj} ( x) $ tend to constants sufficiently rapidly as $ | x | \rightarrow \infty $. In order that the limit-absorption principle holds in this case it is necessary that $ \lambda $ is not an eigen value of $ P $ or that $ f $ 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 $ P = - \Delta + q ( x) $. 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 $ P = ( i \partial / \partial x) $ be an elliptic (or hypo-elliptic) operator satisfying: 1) the polynomial $ P ( \sigma ) $ has real coefficients; 2) the surface $ P ( \sigma ) = 0 $, $ \sigma \in \mathbf R ^ {n} $, decomposes into connected smooth surfaces $ S _ {j} $, $ 1 \leq j \leq k $, whose curvatures do not vanish; and 3) $ \mathop{\rm grad} P ( \sigma ) \neq 0 $ on $ S _ {j} $. Suppose that an orientation is given on $ S _ {j} $, i.e. for each surface one has independently chosen a normal direction $ \nu $. Let $ \omega = x/ | x | $, let $ \sigma _ {j} = \sigma _ {j} ( \omega ) $ be a point on $ S _ {j} $ at which $ \nu $ and $ \omega $ have identical direction and let $ \mu _ {j} ( \omega ) = ( \sigma _ {j} ( \omega ), \omega ) $. Then the function $ u ( x) $ does satisfy the radiation conditions if it can be represented as

$$ u = \sum _ {j = 1 } ^ { k } u _ {j} ( x),\ \ u _ {j} = O ( r ^ {( 1 - n)/2 } ), $$

$$ \frac{\partial u _ {j} }{\partial r } - i \mu _ {j} ( \omega ) u _ {j} = o ( r ^ {( 1 - n)/2 } ),\ r \rightarrow \infty . $$

These conditions determine a unique solution of the equation

$$ P \left ( i { \frac \partial {\partial x } } \right ) u = f,\ \ x \in \mathbf R ^ {n} , $$

for any function $ f $ with compact support. The limit-absorption principle for this equation is that this solution can be obtained as the limit, for $ \epsilon \rightarrow + 0 $, of the unique solution $ u _ \epsilon ( x) \in L _ {2} ( \mathbf R ^ {n} ) $ of the elliptic equation

$$ P \left ( i { \frac \partial {\partial x } } \right ) u _ \epsilon + i \epsilon Q \left ( i { \frac \partial {\partial x } } \right ) u _ \epsilon = f, $$

where $ Q ( \sigma ) $ has real coefficients and $ Q ( \sigma ) \neq 0 $ on $ S _ {j} $. Depending on the choice of $ \mathop{\rm sign} _ {\sigma \in S _ {j} } Q ( \sigma ) $, $ 1 \leq j \leq k $, one obtains in the limit solutions satisfying the radiation conditions corresponding to some orientation of $ S _ {j} $. 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 $ S _ {j} $. 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 $-\Delta u + c u$" Mat. Sb. , 32 : 1 (1953) pp. 109–156 (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.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)