In mathematics, the limiting absorption principle (LAP) is a concept from operator theory and scattering theory that consists of choosing the "correct" resolvent of a linear operator at the essential spectrum based on the behavior of the resolvent near the essential spectrum. The term is often used to indicate that the resolvent, when considered not in the original space (which is usually the L 2 {\displaystyle L^{2}} space), but in certain weighted spaces (usually L s 2 {\displaystyle L_{s}^{2}} , see below), has a limit as the spectral parameter approaches the essential spectrum. This concept developed from the idea of introducing complex parameter into the Helmholtz equation ( Δ + k 2 ) u ( x ) = − F ( x ) {\displaystyle (\Delta +k^{2})u(x)=-F(x)} for selecting a particular solution. This idea is credited to Vladimir Ignatowski, who was considering the propagation and absorption of the electromagnetic waves in a wire. It is closely related to the Sommerfeld radiation condition and the limiting amplitude principle (1948). The terminology – both the limiting absorption principle and the limiting amplitude principle – was introduced by Aleksei Sveshnikov.
Formulation To find which solution to the Helmholz equation with nonzero right-hand side
Δ v ( x ) + k 2 v ( x ) = − F ( x ) , x ∈ R 3 , {\displaystyle \Delta v(x)+k^{2}v(x)=-F(x),\quad x\in \mathbb {R} ^{3},}
with some fixed k > 0 {\displaystyle k>0} , corresponds to the outgoing waves, one considers the limit
v ( x ) = − lim ϵ → + 0 ( Δ + k 2 − i ϵ ) − 1 F ( x ) . {\displaystyle v(x)=-\lim _{\epsilon \to +0}(\Delta +k^{2}-i\epsilon )^{-1}F(x).}
The relation to absorption can be traced to the expression
E ( t , x ) = A e i ( ω t + ϰ x ) {\displaystyle E(t,x)=Ae^{i(\omega t+\varkappa x)}} for the electric field used by Ignatowsky: the absorption corresponds to nonzero imaginary part of ϰ {\displaystyle \varkappa } , and the equation satisfied by E ( t , x ) {\displaystyle E(t,x)} is given by the Helmholtz equation (or reduced wave equation) ( Δ + ϰ 2 / ω 2 ) E ( t , x ) = 0 {\displaystyle (\Delta +\varkappa ^{2}/\omega ^{2})E(t,x)=0} , with
ϰ 2 = μ ε ω 2 c 2 − i 4 π σ μ ω {\displaystyle \varkappa ^{2}={\frac {\mu \varepsilon \omega ^{2}}{c^{2}}}-i4\pi \sigma \mu \omega }
having negative imaginary part (and thus with ϰ 2 / ω 2 {\displaystyle \varkappa ^{2}/\omega ^{2}} no longer belonging to the spectrum of − Δ {\displaystyle -\Delta } ). Above, μ {\displaystyle \mu } is magnetic permeability, σ {\displaystyle \sigma } is electric conductivity, ε {\displaystyle \varepsilon } is dielectric constant, and c {\displaystyle c} is the speed of light in vacuum.
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