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 formula_1 space), but in certain weighted spaces (usually formula_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 formula_3 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
formula_4
with some fixed formula_5, corresponds to the outgoing waves,
one considers the limit
formula_6
The relation to absorption can be traced to the expression
formula_7 for the electric field used by Ignatowsky: the absorption corresponds to nonzero imaginary part of formula_8, and the equation satisfied by formula_9 is given by the Helmholtz equation (or reduced wave equation) formula_10, with
formula_11
having negative imaginary part (and thus with formula_12 no longer belonging to the spectrum of formula_13).
Above, formula_14 is magnetic permeability, formula_15 is electric conductivity, formula_16 is dielectric constant,
and formula_17 is the speed of light in vacuum.
Example and relation to the limiting amplitude principle.
One can consider the Laplace operator in one dimension, which is an unbounded operator formula_18 acting in formula_19 and defined on the domain formula_20, the Sobolev space. Let us describe its resolvent, formula_21. Given the equation
formula_22,
then, for the spectral parameter formula_23 from the resolvent set formula_24, the solution formula_25 is given by
formula_26
where formula_27 is the convolution of F with the fundamental solution G:
formula_28
where the fundamental solution is given by
formula_29
To obtain an operator bounded in formula_19, 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 formula_31 makes sense.
One can also consider the limit of the fundamental solution formula_32 as formula_23 approaches the spectrum of formula_34, given by
formula_35.
Assume that formula_23 approaches formula_37, with some formula_5. 
Depending on whether formula_23 approaches formula_37 in the complex plane from above (formula_41) or from below (formula_42) of the real axis, there will be two different limiting expressions:
formula_43
when formula_44 approaches formula_45 from above and
formula_46
when formula_23 approaches formula_45 from below.
The resolvent formula_49 (convolution with formula_50) corresponds to outgoing waves of the inhomogeneous Helmholtz equation formula_51, while formula_52 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
formula_53
with zero initial data formula_54. A particular solution to the inhomogeneous Helmholtz equation corresponding to outgoing waves is obtained as the limit of formula_55 for large times.
Estimates in the weighted spaces.
Let formula_56 be a linear operator in a Banach space formula_57, defined on the domain formula_58.
For the values of the spectral parameter from the resolvent set of the operator, formula_59, the resolvent formula_21 is bounded when considered as a linear operator acting from formula_57 to itself, formula_62, but its bound depends on the spectral parameter formula_23 and tends to infinity as formula_23 approaches the spectrum of the operator, formula_65. More precisely, there is the relation
formula_66
Many scientists refer to the "limiting absorption principle" when they want to say that the resolvent formula_67 of a particular operator formula_68, when considered as acting in certain weighted spaces, has a limit (and/or remains uniformly bounded) as the spectral parameter formula_23 approaches the essential spectrum, formula_70. 
For instance, in the above example of the Laplace operator in one dimension, formula_71, defined on the domain formula_20, for formula_73, both operators formula_74 with the integral kernels formula_75 are not bounded in formula_1 (that is, as operators from formula_1 to itself), but will both be uniformly bounded when considered as operators
formula_78
with fixed formula_79. The spaces formula_80 are defined as spaces of locally integrable functions such that their formula_2-norm,
formula_82
is finite.
References.
 ;templatestyles src="Reflist/styles.css" / ;