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
is uniquely solvable in , and in certain cases it is possible to find solutions of the equation
by the limit transition
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. ):
The solutions found using this principle are diverging or converging waves and satisfy the radiation conditions at infinity. These results were carried over (cf. , ) 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. ) 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. ). The limit-absorption principle has been substantiated for certain regions with non-compact boundary (cf. , ).
A limit-absorption principle and corresponding radiation conditions have been found for higher-order equations and for systems of equations (cf. –); 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
These conditions determine a unique solution of the equation
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
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. –), as well as in the case of non-convex . For such equations there is also a uniqueness theorem of Kato type.
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Limit-absorption principle. B.R. Vainberg (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Limit-absorption_principle&oldid=13580