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
, 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
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] To find which solution to the Helmholz equation with nonzero right-hand side with some fixed
, corresponds to the outgoing waves, one considers the limit[2][3] The relation to absorption can be traced to the expression
for the electric field used by Ignatowsky: the absorption corresponds to nonzero imaginary part of
, with having negative imaginary part (and thus with
Given the equation then, for the spectral parameter
is the convolution of F with the fundamental solution G: where the fundamental solution is given by To obtain an operator bounded in
, 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
One can also consider the limit of the fundamental solution
) of the real axis, there will be two different limiting expressions:
) corresponds to outgoing waves of the inhomogeneous Helmholtz equation
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
A particular solution to the inhomogeneous Helmholtz equation corresponding to outgoing waves is obtained as the limit of
be a linear operator in a Banach space
For the values of the spectral parameter from the resolvent set of the operator,
is bounded when considered as a linear operator acting from
, but its bound depends on the spectral parameter
More precisely, there is the relation Many scientists refer to the "limiting absorption principle" when they want to say that the resolvent
, when considered as acting in certain weighted spaces, has a limit (and/or remains uniformly bounded) as the spectral parameter
approaches the essential spectrum,
For instance, in the above example of the Laplace operator in one dimension,
to itself), but will both be uniformly bounded when considered as operators with fixed
are defined as spaces of locally integrable functions such that their