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

References

1. Tikhonov, A.N. and Samarskii, A.A. (1948). "On the radiation principle" (PDF). Zh. Eksper. Teoret. Fiz. 18 (2): 243–248.{{cite journal}}: CS1 maint: multiple names: authors list (link)
2. Sveshnikov, A.G. (1950). "Radiation principle". Doklady Akademii Nauk SSSR. Novaya Seriya. 5: 917–920. Zbl 0040.41903.
3. Smirnov, V.I. (1974). Course in Higher Mathematics. Vol. 4 (6 ed.). Moscow, Nauka.