Limiting amplitude principle

{{Short description|Mathematical concept for solving the Helmholtz equation}}

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.{{ cite journal

|author=Tikhonov, A.N. and Samarskii, A.A.

|title=On the radiation principle

|journal=Zh. Eksper. Teoret. Fiz.

|url=http://samarskii.ru/articles/1948/1948_003ocr.pdf

|pages=243–248

|volume=18

|number=2

|year=1948

}}

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.{{ cite journal

|author=Sveshnikov, A.G.

|title=Radiation principle

|journal=Doklady Akademii Nauk SSSR |series=Novaya Seriya

|url=http://mi.mathnet.ru/eng/dan50544

|pages=917–920

|volume=5

|year=1950|zbl=0040.41903}}

Formulation

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

: $\Delta v(x)+k^2 v(x)=-F(x),\quad x\in\R^3,$

with some fixed $k>0$ , corresponds to the outgoing waves,

one considers the wave equation with the source term,

: $(\Delta-\partial_t^2)u(x,t)=-F(x)e^{-i k t},\quad t\ge 0, \quad x\in\R^3,$

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

: $v(x)=\lim_{t\to +\infty}u(x,t)e^{i k t}$

for large times.{{ cite book

|author=Smirnov, V.I.

|title=Course in Higher Mathematics

|volume=4

|publisher=Moscow, Nauka

|year=1974

|edition=6

|url=http://edu.sernam.ru/book_sm_math42.php?id=133

}}