Limiting amplitude principle

In mathematics, the limiting amplitude principle is a concept from operator theory and scattering theory used for choosing a particular solution to the Helmholtz equation. The choice is made by considering a particular time-dependent problem of the forced oscillations due to the action of a periodic force. The principle was introduced by Andrey Nikolayevich Tikhonov and Alexander Andreevich Samarskii.^[1] It is closely related to the limiting absorption principle (1905) and the Sommerfeld radiation condition (1912). The terminology -- both the limiting absorption principle and the limiting amplitude principle -- was introduced by Aleksei Sveshnikov.^[2]

Formulation

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 wave equation with the source term,

${\displaystyle (\Delta -\partial _{t}^{2})u(x,t)=-F(x)e^{-ikt},\quad t\geq 0,\quad x\in \mathbb {R} ^{3},}$

with zero initial data ${\displaystyle u(x,0)=0,\,\partial _{t}u(x,0)=0}$. A particular solution to the Helmholtz equation corresponding to outgoing waves is obtained as the limit

${\displaystyle v(x)=\lim _{t\to +\infty }u(x,t)e^{ikt}}$

for large times.^[1][3]

See also

• Limiting absorption principle
• Sommerfeld radiation condition