Limiting absorption principle

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 particular solutions. This idea is credited to Vladimir Ignatowski, who was considering the propagation and absorption of the electromagnetic waves in a wire.
The relation with absorption can be traced to the expression
formula_4 for the electric field used by Ignatowsky: the absorption corresponds to nonzero imaginary part of formula_5, and the equation satisfied by formula_6 is given by the Helmholtz equation (or reduced wave equation) formula_7, with formula_8 having nonzero imaginary part (and thus no longer belonging to the spectrum of formula_9).
Example and relation to the limiting amplitude principle.
One can consider the Laplace operator in one dimension, which is an unbounded operator formula_10 acting in formula_11 and defined on the domain formula_12, the Sobolev space. Let us describe its resolvent, formula_13. Given the equation
then, for the spectral parameter formula_15 from the resolvent set formula_16, the solution formula_17 is given by
formula_18
where formula_19 is the convolution of with the fundamental solution :
where the fundamental solution is given by
To obtain an operator bounded in formula_11, one needs to use the branch of the square root which has positive real part (which decays for large absolute value of ), so that the convolution of with formula_23 makes sense.
One can also consider the limit of the fundamental solution formula_24 as formula_15 approaches the spectrum of formula_26, given by
formula_27.
Assume that formula_15 approaches formula_29, with some formula_30. 
Depending on whether formula_15 approaches formula_29 in the complex plane from above (formula_33) or from below (formula_34) of the real axis, there will be two different limiting expressions:
formula_35
when formula_36 approaches formula_37 from above and
formula_38
when formula_15 approaches formula_40 from below.
The resolvent formula_41 (convolution with formula_42) corresponds to outgoing waves of the inhomogeneous Helmholtz equation formula_43, while formula_44 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
with zero initial data formula_46. A particular solution to the inhomogeneous Helmholtz equation corresponding to outgoing waves is obtained as the limit of formula_47 for large times.
Estimates in the weighted spaces.
Let formula_48 be a linear operator in a Banach space formula_49, defined on the domain formula_50.
For the values of the spectral parameter from the resolvent set of the operator, formula_51, the resolvent formula_13 is bounded when considered as a linear operator acting from formula_49 to itself, formula_54, but its bound depends on the spectral parameter formula_15 and tends to infinity as formula_15 approaches the spectrum of the operator, formula_57. More precisely, there is the relation
Many scientists refer to the "limiting absorption principle" when they want to say that the resolvent formula_59 of a particular operator , when considered as acting in certain weighted spaces, has a limit (and/or remains uniformly bounded) as the spectral parameter formula_15 approaches the essential spectrum, formula_61. 
For instance, in the above example of the Laplace operator in one dimension, formula_62, defined on the domain formula_12, for formula_64, both operators formula_65 with the integral kernels formula_66 are not bounded in formula_1 (that is, as operators from formula_1 to itself), but will both be bounded when considered as operators
where the spaces formula_70 are defined as spaces of locally integrable functions such that their formula_2-norm,
is finite.