Limiting absorption principle

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]

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]