In
mathematical physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
, more specifically the one-dimensional
inverse scattering problem
In mathematics and physics, the inverse scattering problem is the problem of determining characteristics of an object, based on data of how it scatters incoming radiation or particles. It is the inverse problem to the direct scattering problem, wh ...
, the Marchenko equation (or Gelfand-Levitan-Marchenko equation or GLM equation), named after
Israel Gelfand
Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand ( yi, ישראל געלפֿאַנד, russian: Изра́иль Моисе́евич Гельфа́нд, uk, Ізраїль Мойсейович Гел ...
,
Boris Levitan
Boris Levitan (7 June 1914 – 4 April 2004) was a mathematician known in particular for his work on almost periodic functions, and Sturm–Liouville operators, especially, on inverse scattering.
Life
Boris Levitan was born in Berdyansk ...
and
Vladimir Marchenko
Vladimir Alexandrovich Marchenko (russian: Влади́мир Алекса́ндрович Ма́рченко, uk, Володи́мир Олекса́ндрович Ма́рченко; born 7 July 1922) is a Soviet and Ukrainian mathematician who ...
, is derived by computing the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
of the scattering relation:
:
Where
is a
symmetric kernel
In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in ...
, such that
which is computed from the scattering data. Solving the Marchenko equation, one obtains the kernel of the transformation operator
from which the potential can be read off. This equation is derived from the
Gelfand–Levitan integral equation, using the
Povzner–Levitan representation.
See also
*
Lax pair In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the ''Lax equation''. Lax pairs were introduced by Peter Lax to discuss sol ...
References
*
Integral equations
Scattering theory
{{scattering-stub