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

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

with some fixed , corresponds to the outgoing waves, one considers the wave equation with the source term,

with zero initial data . A particular solution to the Helmholtz equation corresponding to outgoing waves is obtained as the limit

for large times.[1][3]

See also

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References

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  1. ^ a b 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.