Fredholm determinants and inverse scattering problems

@article{Dyson1976FredholmDA,
  title={Fredholm determinants and inverse scattering problems},
  author={Freeman J. Dyson},
  journal={Communications in Mathematical Physics},
  year={1976},
  volume={47},
  pages={171-183}
}
  • F. Dyson
  • Published 1 June 1976
  • Mathematics
  • Communications in Mathematical Physics
The Gel'fand-Levitan and Marchenko formalisms for solving the inverse scattering problem are applied together to a single set of scattering phase-shifts. The result is an identity relating two different types of Fredholm determinant. As an application of the method, an asymptotic formula of high accuracy is derived for a particular Fredholm determinant that determines the level-spacing distribution-function in the theory of random matrices. 
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References

SHOWING 1-10 OF 16 REFERENCES
Toeplitz Forms And Their Applications
Part I: Toeplitz Forms: Preliminaries Orthogonal polynomials. Algebraic properties Orthogonal polynomials. Limit properties The trigonometric moment problem Eigenvalues of Toeplitz forms
Asymptotic behavior of spacing distributions for the eigenvalues of random matrices
It is known that the probability Eβ(0, S) that an arbitrary interval of length S contains none of the eigenvalues of a random matrix chosen from the orthogonal (β = 1), unitary (β = 2) or symplectic
Statistical Theory of the Energy Levels of Complex Systems. I
New kinds of statistical ensemble are defined, representing a mathematical idealization of the notion of ``all physical systems with equal probability.'' Three such ensembles are studied in detail,
Dokl
  • Akad. Nauk SSSR 72, 457—460 (1950); 104, 695—698
  • 1955
Izv
  • Akad. Nauk SSSR, Ser. Mat. 15, 309—360 (1951); English translation in Am. Math. Soc. Translations (2) 1, 253—304
  • 1955
Usp . Mat . Nauk 14 , Part 4 , 57 — 119 ( 1959 ) ; English translation in
  • Dokl . Akad . Nauk SSSR
...
...