In mathematical physics, more specifically the one-dimensional inverse scattering problem, the Marchenko equation (or Gelfand-Levitan-Marchenko equation or GLM equation), named after Israel Gelfand, Boris Levitan and Vladimir Marchenko, is derived by computing the Fourier transform of the scattering relation:
Where is a symmetric kernel, 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.
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Transcription
Application to scattering theory
Suppose that for a potential for the Schrödinger operator , one has the scattering data , where are the reflection coefficients from continuous scattering, given as a function , and the real parameters are from the discrete bound spectrum.[1]
Then defining
See also
References
- ^ Dunajski, Maciej (2015). Solitons, instantons, and twistors (1. publ., corrected 2015 ed.). Oxford: Oxford University Press. ISBN 978-0198570639.
- Marchenko, V. A. (2011). Sturm–Liouville Operators and Applications (2nd ed.). Providence: American Mathematical Society. ISBN 978-0-8218-5316-0. MR 2798059.
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