In mathematics, the limiting absorption principle 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, but in certain weighted spaces, has a limit as the spectral parameter approaches the essential spectrum. This concept developed from the idea of introducing small absorption into the wave equation for selecting particular solutions, which goes back to Vladimir Ignatowski.
As an example, let us consider the Laplace operator in one dimension, which is an unbounded operatoracting in and defined on the domain, the Sobolev space. Let us describe its resolvent,. Given the equation then, for the spectral parameter from the resolvent set, the solution is given by where is the convolution of with the fundamental solution : with the fundamental solution given by It is clear which of the branches of the square root one needs to pick: the one with positive real part, so that the convolution of with makes sense. One can consider the limit of the fundamental solution as approaches the spectrum of, given by Depending on whether approaches the spectrum from above or from below, there will be two different limiting expressions: if and . What do these two different limits correspond to? Let us recall that one arrives at the above spectral problem when studying the Schrödinger equation, The word "absorption" is due to the fact that if the medium were absorbing, then the equation would be , the solution with would gain a temporal decay:, ; the "limiting absorption" means that this imaginary part tends to zero. Due to this decay in time for positive times, the Fourier transform in time of the solution, could be analytically extended into a small region of the lower half-plane,, with. In this sense, the "correct" resolvent, the one corresponding to the outgoing waves, would be represented by the operator with the integral kernel, which is defined as the limit of the resolvent when approaching the spectrum from the region.
Estimates in the weighted spaces
Let be a linear operator in a Banach space, defined on the domain. For the values of the spectral parameter from the resolvent set of the operator,, the resolvent is bounded when considered as a linear operator acting from to itself,, but its bound depends on the spectral parameter and tends to infinity as approaches the spectrum of the operator,. More precisely, there is the relation In recent years, many scientists refer to the "limiting absorption principle" when they want to say that the resolvent of a particular operator, when considered as acting in certain weighted spaces, has a limit as the spectral parameter approaches the essential spectrum,. For instance, in the above example of the Laplace operator in one dimension,, defined on the domain, for, both operators with the integral kernels are not bounded in , but will both be bounded when considered as operators where the spaces are defined as spaces of locally integrable functions such that their -norm, is finite.
