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 ${\displaystyle L^{2}}$ 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]