Limiting absorption principle

To find which solution to the Helmholz equation with nonzero right-hand side

${\displaystyle \Delta v(x)+k^{2}v(x)=-F(x),\quad x\in \mathbb {R} ^{3},}$

with some fixed ${\displaystyle k>0}$, corresponds to the outgoing waves, one considers the limit[2][3]

${\displaystyle v(x)=-\lim _{\epsilon \to +0}(\Delta +k^{2}-i\epsilon )^{-1}F(x).}$

The relation to absorption can be traced to the expression ${\displaystyle E(t,x)=Ae^{i(\omega t+\varkappa x)}}$ for the electric field used by Ignatowsky: the absorption corresponds to nonzero imaginary part of ${\displaystyle \varkappa }$, and the equation satisfied by ${\displaystyle E(t,x)}$ is given by the Helmholtz equation (or reduced wave equation) ${\displaystyle (\Delta +\varkappa ^{2}/\omega ^{2})E(t,x)=0}$, with

${\displaystyle \varkappa ^{2}={\frac {\mu \varepsilon \omega ^{2}}{c^{2}}}-i4\pi \sigma \mu \omega }$

having negative imaginary part (and thus with ${\displaystyle \varkappa ^{2}/\omega ^{2}}$ no longer belonging to the spectrum of ${\displaystyle -\Delta }$). Above, ${\displaystyle \mu }$ is magnetic permeability, ${\displaystyle \sigma }$ is electric conductivity, ${\displaystyle \varepsilon }$ is dielectric constant, and ${\displaystyle c}$ is the speed of light in vacuum.[1]

One can 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, ${\displaystyle R(z)=(A-zI)^{-1}}$. Given the equation

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

then, for the spectral parameter ${\displaystyle z}$ from the resolvent set ${\displaystyle \mathbb {C} \setminus [0,+\infty )}$, the solution ${\displaystyle u\in L^{2}(\mathbb {R} )}$ is given by ${\displaystyle u(x)=(R(z)F)(x)=(G(\cdot ,z)*F)(x),}$ where ${\displaystyle G(\cdot ,z)*F}$ is the convolution of F with the fundamental solution G:

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

where the fundamental solution is given by

${\displaystyle G(x;z)={\frac {1}{2{\sqrt {-z}}}}e^{-|x|{\sqrt {-z}}},\quad z\in \mathbb {C} \setminus [0,+\infty ).}$

To obtain an operator bounded in ${\displaystyle L^{2}(\mathbb {R} )}$, 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 ${\displaystyle F\in L^{2}(\mathbb {R} )}$ makes sense.

One can also consider the limit of the fundamental solution ${\displaystyle G(x;z)}$ as ${\displaystyle z}$ approaches the spectrum of ${\displaystyle -\partial _{x}^{2}}$, given by ${\displaystyle \sigma (-\partial _{x}^{2})=[0,+\infty )}$. Assume that ${\displaystyle z}$ approaches ${\displaystyle k^{2}}$, with some ${\displaystyle k>0}$. Depending on whether ${\displaystyle z}$ approaches ${\displaystyle k^{2}}$ in the complex plane from above (${\displaystyle \Im (z)>0}$) or from below (${\displaystyle \Im (z)<0}$) of the real axis, there will be two different limiting expressions: ${\displaystyle G_{+}(x;k^{2})=\lim _{\varepsilon \to 0+}G(x;k^{2}+i\varepsilon )=-{\frac {1}{2ik}}e^{i|x|k}}$ when ${\displaystyle z\in \mathbb {C} }$ approaches ${\displaystyle k^{2}\in (0,+\infty )}$ from above and ${\displaystyle G_{-}(x;k^{2})=\lim _{\varepsilon \to 0+}G(x;k^{2}-i\varepsilon )={\frac {1}{2ik}}e^{-i|x|k}}$ when ${\displaystyle z}$ approaches ${\displaystyle k^{2}\in (0,+\infty )}$ from below. The resolvent ${\displaystyle R_{+}(k^{2})}$ (convolution with ${\displaystyle G_{+}(x;k^{2})}$) corresponds to outgoing waves of the inhomogeneous Helmholtz equation ${\displaystyle (-\partial _{x}^{2}-k^{2})u(x)=F(x)}$, while ${\displaystyle R_{-}(k^{2})}$ 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

${\displaystyle (\partial _{t}^{2}-\partial _{x}^{2})\psi (t,x)=F(x)e^{-ikt},\quad t\geq 0,\quad x\in \mathbb {R} ,}$

with zero initial data ${\displaystyle \psi (0,x)=0,\,\partial _{t}\psi (t,x)|_{t=0}=0}$. A particular solution to the inhomogeneous Helmholtz equation corresponding to outgoing waves is obtained as the limit of ${\displaystyle \psi (t,x)e^{ikt}}$ for large times.[3]

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, ${\displaystyle z\in \rho (A)\subset \mathbb {C} }$, the resolvent ${\displaystyle R(z)=(A-zI)^{-1}}$ is bounded when considered as a linear operator acting from ${\displaystyle X}$ to itself, ${\displaystyle R(z):\,X\to X}$, but its bound depends on the spectral parameter ${\displaystyle z}$ and tends to infinity as ${\displaystyle z}$ approaches the spectrum of the operator, ${\displaystyle \sigma (A)=\mathbb {C} \setminus \rho (A)}$. More precisely, there is the relation

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

Many scientists refer to the "limiting absorption principle" when they want to say that the resolvent ${\displaystyle R(z)}$ of a particular operator ${\displaystyle A}$, when considered as acting in certain weighted spaces, has a limit (and/or remains uniformly bounded) as the spectral parameter ${\displaystyle z}$ approaches the essential spectrum, ${\displaystyle \sigma _{\mathrm {ess} }(A)}$. 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 ${\displaystyle z>0}$, both operators ${\displaystyle R_{\pm }(z)}$ with the integral kernels ${\displaystyle G_{\pm }(x-y;z)}$ are not bounded in ${\displaystyle L^{2}}$ (that is, as operators from ${\displaystyle L^{2}}$ to itself), but will both be uniformly bounded when considered as operators

${\displaystyle R_{\pm }(z):\;L_{s}^{2}(\mathbb {R} )\to L_{-s}^{2}(\mathbb {R} ),\quad s>1/2,\quad z\in \mathbb {C} \setminus [0,+\infty ),\quad |z|\geq \delta ,}$

with fixed ${\displaystyle \delta >0}$. 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.[4][5]

• Sommerfeld radiation condition
• Limiting amplitude principle