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A manner for uniquely finding solutions to equations analogous to the [[Helmholtz equation|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. [[#References|[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|radiation conditions]] at infinity. These results were carried over (cf. [[#References|[2]]], [[#References|[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. [[#References|[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. [[#References|[3]]]). The limit-absorption principle has been substantiated for certain regions with non-compact boundary (cf. [[#References|[3]]], [[#References|[4]]]).

A limit-absorption principle and corresponding radiation conditions have been found for higher-order equations and for systems of equations (cf. [[#References|[5]]]–[[#References|[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. [[#References|[5]]]–[[#References|[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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. [V.S. Ignatovskii] Ignatowsky,   "Reflexion elektromagnetischer Wellen an einem Drahte"  ''Ann. der Physik'' , '''18''' :  13  (1905)  pp. 495–522</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</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.V. Sveshnikov,   "The limit absorption principle for a wave guide"  ''Dokl. Akad. Nauk SSSR'' , '''80''' :  3  (1951)  pp. 345–347  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  B.R. Vainberg,   "Asymptotic methods in equations of mathematical physics" , Gordon &amp; Breach  (1988)  (Translated from Russian)</TD></TR></table>