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 small absorption into the wave equation for selecting particular solutions, which goes back to Vladimir Ignatowski.[1]

Relation to the scattering theory

As an example, let us consider the Laplace operator in one dimension, which is an unbounded operator ${\displaystyle A=-\partial _{x}^{2},}$ acting in ${\displaystyle L^{2}(\mathbb {R} )}$ and defined on the domain ${\displaystyle D(A)=H^{2}(\mathbb {R} )}$, the Sobolev space. Let us describe its resolvent, ${\displaystyle R(\lambda )=(A-\lambda I)^{-1}}$. Given the equation

${\displaystyle (-\partial _{x}^{2}-\lambda )u(x)=f(x),\quad x\in \mathbb {R} ,\quad f\in L^{2}(\mathbb {R} )}$,

then, for the spectral parameter ${\displaystyle \lambda }$ from the resolvent set ${\displaystyle \mathbb {C} \smallsetminus {\overline {\mathbb {R} _{+}}}}$, the solution ${\displaystyle u\in L^{2}(\mathbb {R} )}$ is given by ${\displaystyle u(x)=(R(\lambda )f)(x)=(G(\cdot ,\lambda )*f)(x),}$ where ${\displaystyle G(\cdot ,\lambda )*f}$ is the convolution of f with the fundamental solution G:

${\displaystyle (G(\cdot ,\lambda )*f)(x)=\int _{\mathbb {R} }G(x-y;\lambda )f(y)\,dy,}$

with the fundamental solution given by

${\displaystyle G(x;\lambda )={\frac {1}{2{\sqrt {-\lambda }}}}e^{-|x|{\sqrt {-\lambda }}},\quad \lambda \in \mathbb {C} \smallsetminus {\overline {\mathbb {R} _{+}}}.}$

It is clear which of the branches of the square root one needs to pick: the one with positive real part (which decays for large absolute value of x), so that the convolution of G with ${\displaystyle f\in L^{2}(\mathbb {R} )}$ makes sense.

One can consider the limit of the fundamental solution ${\displaystyle G(x;\lambda )}$ as ${\displaystyle \lambda }$ approaches the spectrum of ${\displaystyle -\partial _{x}^{2}}$, given by ${\displaystyle \sigma (-\partial _{x}^{2})={\overline {\mathbb {R} _{+}}}=[0,+\infty )}$. Depending on whether ${\displaystyle \lambda }$ approaches the spectrum from above or from below, there will be two different limiting expressions: ${\displaystyle G_{+}(x;\lambda )=\lim _{\varepsilon \to 0+}G(x;\lambda +i\varepsilon )=-{\frac {1}{2i{\sqrt {\lambda }}}}e^{i|x|{\sqrt {\lambda }}}}$ if ${\displaystyle \lambda =|\lambda |+i0}$ (when ${\displaystyle \lambda }$ approaches ${\displaystyle \mathbb {R} _{+}}$ from above) and ${\displaystyle G_{-}(x;\lambda )=\lim _{\varepsilon \to 0+}G(x;\lambda -i\varepsilon )={\frac {1}{2i{\sqrt {\lambda }}}}e^{-i|x|{\sqrt {\lambda }}}}$ (when approaching ${\displaystyle \mathbb {R} _{+}}$ from below).

What do these two different limits correspond to? Let us recall that one arrives at the above spectral problem when studying the Schrödinger equation,

${\displaystyle i\,\partial _{t}\psi (t,x)=-\partial _{x}^{2}\psi (t,x),\quad t\in \mathbb {R} ,\quad x\in \mathbb {R} .}$

The word "absorption" is due to the fact that if the medium were absorbing, then the equation would be ${\displaystyle i\,\partial _{t}\psi (t,x)=-\partial _{x}^{2}\psi (t,x)-i\varepsilon \psi (t,x)}$, the solution with ${\displaystyle \varepsilon >0}$ would gain a temporal decay: ${\displaystyle \psi (t,x)\to \psi (t,x)e^{-\varepsilon t}}$, ${\displaystyle t\to +\infty }$; 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,

${\displaystyle {\hat {\psi }}(\lambda ,x)=\int _{\mathbb {R} }e^{i\lambda t}\psi (t,x)\,dt,}$

could be analytically extended into a small region of the lower half-plane, ${\displaystyle \lambda \in \mathbb {C} }$, with ${\displaystyle \Im \lambda >-\varepsilon }$. In this sense, the "correct" resolvent, the one corresponding to the outgoing waves, would be represented by the operator ${\displaystyle R_{-}(\lambda )}$ with the integral kernel ${\displaystyle G_{-}(x-y,\lambda )}$, which is defined as the limit of the resolvent when approaching the spectrum from the region ${\displaystyle \Im \lambda <0}$.[2]

Estimates in the weighted spaces

Let ${\displaystyle A:\,X\to X}$ be a linear operator in a Banach space ${\displaystyle X}$, defined on the domain ${\displaystyle D(A)\subset X}$. For the values of the spectral parameter from the resolvent set of the operator, ${\displaystyle \lambda \in \rho (A)\subset \mathbb {C} }$, the resolvent ${\displaystyle R(\lambda )=(A-\lambda I)^{-1}}$ is bounded when considered as a linear operator acting from ${\displaystyle X}$ to itself, ${\displaystyle R(\lambda ):\,X\to X}$, but its bound depends on the spectral parameter ${\displaystyle \lambda }$ and tends to infinity as ${\displaystyle \lambda }$ approaches the spectrum of the operator, ${\displaystyle \sigma (A)=\mathbb {C} \smallsetminus \rho (A)}$. More precisely, there is the relation

${\displaystyle \Vert R(\lambda )\Vert \geq {\frac {1}{\operatorname {dist} (\lambda ,\sigma (A))}},\qquad \lambda \in \rho (A).}$

In recent years, many scientists refer to the "limiting absorption principle" when they want to say that the resolvent ${\displaystyle R(\lambda )}$ of a particular operator A, when considered as acting in certain weighted spaces, has a limit (and/or remains uniformly bounded) as the spectral parameter ${\displaystyle \lambda }$ approaches the essential spectrum, ${\displaystyle \sigma _{ess}(A)}$. For instance, in the above example of the Laplace operator in one dimension, ${\displaystyle A=-\partial _{x}^{2}:\,L^{2}(\mathbb {R} )\to L^{2}(\mathbb {R} )}$, defined on the domain ${\displaystyle D(A)=H^{2}(\mathbb {R} )}$, for ${\displaystyle \lambda >0}$, both operators ${\displaystyle R_{\pm }(\lambda )}$ with the integral kernels ${\displaystyle G_{\pm }(x-y;\lambda )}$ are not bounded in ${\displaystyle L^{2}}$ (that is, as operators from ${\displaystyle L^{2}}$ to itself), but will both be bounded when considered as operators

${\displaystyle R_{\pm }(\lambda ):\;L_{s}^{2}(\mathbb {R} )\to L_{-s}^{2}(\mathbb {R} ),\quad s>1/2,\quad \lambda \in \mathbb {C} \smallsetminus {\overline {\mathbb {R} _{+}}},}$

where the spaces ${\displaystyle L_{s}^{2}(\mathbb {R} )}$ are defined as spaces of locally integrable functions such that their ${\displaystyle L_{s}^{2}}$-norm,

${\displaystyle \Vert u\Vert _{L_{s}^{2}(\mathbb {R} )}^{2}=\int _{\mathbb {R} }(1+x^{2})^{s}|u(x)|^{2}\,dx,}$

is finite.[3][4]

References

1. W. v. Ignatowsky (1905). "Reflexion elektromagnetischer Wellen an einem Draft". Annalen der Physik. 18: 495–522.
2. Smirnov, V.I. (1974). Course in Higher Mathematics. 4 (6 ed.). Moscow, Nauka.
3. Agmon, S (1975). "Spectral properties of Schrödinger operators and scattering theory," (PDF). Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4). 2: 151–218.
4. Reed, Michael C.; Simon, Barry (1978). Methods of modern mathematical physics. Analysis of operators. 4. Academic Press. ISBN 0-12-585004-2.