Limiting absorption principle – Wikipedia

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

${displaystyle L^{2}}$

space), but in certain weighted spaces (usually

${displaystyle L_{s}^{2}}$

, 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

${displaystyle (Delta +k^{2})u(x)=-F(x)}$

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.^[1]
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.^[2]

Formulation[edit]

To find which solution to the Helmholz equation with nonzero right-hand side

${displaystyle Delta v(x)+k^{2}v(x)=-F(x),quad xin mathbb {R} ^{3},}$

with some fixed

${displaystyle k>0}$

[2]^[3]

${displaystyle v(x)=-lim _{epsilon to +0}(Delta +k^{2}-iepsilon )^{-1}F(x).}$

The relation to absorption can be traced to the expression

${displaystyle E(t,x)=Ae^{i(omega t+varkappa x)}}$

for the electric field used by Ignatowsky: the absorption corresponds to nonzero imaginary part of

${displaystyle varkappa }$

, and the equation satisfied by

${displaystyle E(t,x)}$

is given by the Helmholtz equation (or reduced wave equation)

${displaystyle (Delta +varkappa ^{2}/omega ^{2})E(t,x)=0}$

, with

${displaystyle varkappa ^{2}={frac {mu varepsilon omega ^{2}}{c^{2}}}-i4pi sigma mu omega }$

having negative imaginary part (and thus with

${displaystyle varkappa ^{2}/omega ^{2}}$

no longer belonging to the spectrum of

${displaystyle -Delta }$

).
Above,

${displaystyle mu }$

is magnetic permeability,

${displaystyle sigma }$

is electric conductivity,

${displaystyle varepsilon }$

is dielectric constant,
and

${displaystyle c}$

is the speed of light in vacuum.^[1]

Example and relation to the limiting amplitude principle[edit]

One can consider the Laplace operator in one dimension, which is an unbounded operator

${displaystyle A=-partial _{x}^{2},}$

acting in

${displaystyle L^{2}(mathbb {R} )}$

and defined on the domain

${displaystyle D(A)=H^{2}(mathbb {R} )}$

, the Sobolev space. Let us describe its resolvent,

${displaystyle R(z)=(A-zI)^{-1}}$

. Given the equation

${displaystyle (-partial _{x}^{2}-z)u(x)=F(x),quad xin mathbb {R} ,quad Fin L^{2}(mathbb {R} )}$

,

then, for the spectral parameter

${displaystyle z}$

from the resolvent set

${displaystyle mathbb {C} setminus [0,+infty )}$

, the solution

${displaystyle uin L^{2}(mathbb {R} )}$

is given by

${displaystyle u(x)=(R(z)F)(x)=(G(cdot ,z)*F)(x),}$

where

${displaystyle G(cdot ,z)*F}$

is the convolution of F with the fundamental solution G:

${displaystyle (G(cdot ,z)*F)(x)=int _{mathbb {R} }G(x-y;z)F(y),dy,}$

where the fundamental solution is given by

${displaystyle G(x;z)={frac {1}{2{sqrt {-z}}}}e^{-|x|{sqrt {-z}}},quad zin mathbb {C} setminus [0,+infty ).}$

To obtain an operator bounded in

${displaystyle L^{2}(mathbb {R} )}$

, one needs to use the branch of the square root which has positive real part (which decays for large absolute value of x), so that the convolution of G with

${displaystyle Fin L^{2}(mathbb {R} )}$

makes sense.

One can also consider the limit of the fundamental solution

${displaystyle G(x;z)}$

as

${displaystyle z}$

approaches the spectrum of

${displaystyle -partial _{x}^{2}}$

, given by

${displaystyle sigma (-partial _{x}^{2})=[0,+infty )}$

.
Assume that

${displaystyle z}$

approaches

${displaystyle k^{2}}$

, with some

${displaystyle k>0}$

${displaystyle z}$

approaches

${displaystyle k^{2}}$

in the complex plane from above (

${displaystyle Im (z)>0}$

${displaystyle Im (z)<0}$

) of the real axis, there will be two different limiting expressions:

${displaystyle G_{+}(x;k^{2})=lim _{varepsilon to 0+}G(x;k^{2}+ivarepsilon )=-{frac {1}{2ik}}e^{i|x|k}}$

when

${displaystyle zin mathbb {C} }$

approaches

${displaystyle k^{2}in (0,+infty )}$

from above and

${displaystyle G_{-}(x;k^{2})=lim _{varepsilon to 0+}G(x;k^{2}-ivarepsilon )={frac {1}{2ik}}e^{-i|x|k}}$

when

${displaystyle z}$

approaches

${displaystyle k^{2}in (0,+infty )}$

from below.
The resolvent

${displaystyle R_{+}(k^{2})}$

(convolution with

${displaystyle G_{+}(x;k^{2})}$

) corresponds to outgoing waves of the inhomogeneous Helmholtz equation

${displaystyle (-partial _{x}^{2}-k^{2})u(x)=F(x)}$

, while

${displaystyle R_{-}(k^{2})}$

corresponds to incoming waves.
This is directly related to the limiting amplitude principle:
to find which solution corresponds to the outgoing waves,
one considers the inhomogeneous wave equation

${displaystyle (partial _{t}^{2}-partial _{x}^{2})psi (t,x)=F(x)e^{-ikt},quad tgeq 0,quad xin mathbb {R} ,}$

with zero initial data

${displaystyle psi (0,x)=0,,partial _{t}psi (t,x)|_{t=0}=0}$

. A particular solution to the inhomogeneous Helmholtz equation corresponding to outgoing waves is obtained as the limit of

${displaystyle psi (t,x)e^{ikt}}$

for large times.^[3]

Estimates in the weighted spaces[edit]

Let

${displaystyle A:,Xto X}$

be a linear operator in a Banach space

${displaystyle X}$

, defined on the domain

${displaystyle D(A)subset X}$

.
For the values of the spectral parameter from the resolvent set of the operator,

${displaystyle zin rho (A)subset mathbb {C} }$

, the resolvent

${displaystyle R(z)=(A-zI)^{-1}}$

is bounded when considered as a linear operator acting from

${displaystyle X}$

to itself,

${displaystyle R(z):,Xto X}$

, but its bound depends on the spectral parameter

${displaystyle z}$

and tends to infinity as

${displaystyle z}$

approaches the spectrum of the operator,

${displaystyle sigma (A)=mathbb {C} setminus rho (A)}$

. More precisely, there is the relation

${displaystyle Vert R(z)Vert geq {frac {1}{operatorname {dist} (z,sigma (A))}},qquad zin rho (A).}$

Many scientists refer to the “limiting absorption principle” when they want to say that the resolvent

${displaystyle R(z)}$

of a particular operator

${displaystyle A}$

, when considered as acting in certain weighted spaces, has a limit (and/or remains uniformly bounded) as the spectral parameter

${displaystyle z}$

approaches the essential spectrum,

${displaystyle sigma _{mathrm {ess} }(A)}$

.
For instance, in the above example of the Laplace operator in one dimension,

${displaystyle A=-partial _{x}^{2}:,L^{2}(mathbb {R} )to L^{2}(mathbb {R} )}$

, defined on the domain

${displaystyle D(A)=H^{2}(mathbb {R} )}$

, for

${displaystyle z>0}$

${displaystyle R_{pm }(z)}$

with the integral kernels

${displaystyle G_{pm }(x-y;z)}$

are not bounded in

${displaystyle L^{2}}$

(that is, as operators from

${displaystyle L^{2}}$

to itself), but will both be uniformly bounded when considered as operators

${displaystyle R_{pm }(z):;L_{s}^{2}(mathbb {R} )to L_{-s}^{2}(mathbb {R} ),quad s>1/2,quad zin mathbb {C} setminus [0,+infty ),quad |z|geq delta ,}$

${displaystyle delta >0}$

${displaystyle L_{s}^{2}(mathbb {R} )}$

are defined as spaces of locally integrable functions such that their

${displaystyle L_{s}^{2}}$

-norm,

${displaystyle Vert uVert _{L_{s}^{2}(mathbb {R} )}^{2}=int _{mathbb {R} }(1+x^{2})^{s}|u(x)|^{2},dx,}$

is finite.^[4][5]

See also[edit]

References[edit]