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 " 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)
How to Cite This Entry:
Limit-absorption principle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Limit-absorption_principle&oldid=47637
This article was adapted from an original article by B.R. Vainberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article