×

Fredholm determinants and inverse scattering problems. (English) Zbl 0323.33008

Summary: 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.

MSC:

81U40 Inverse scattering problems in quantum theory
47A40 Scattering theory of linear operators
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Gel’fand, I.M., Levitan, B.M.: Izv. Akad. Nauk SSSR, Ser. Mat.15, 309–360 (1951); English translation in Am. Math. Soc. Translations (2)1, 253–304 (1955)
[2] Marchenko, V.A.: Dokl. Akad. Nauk SSSR72, 457–460 (1950);104, 695–698 (1955)
[3] Faddeev, L.D.: Usp. Mat. Nauk14, Part 4, 57–119 (1959); English translation in J. Math. Phys.4, 72–104 (1963)
[4] Mehta, M.L.: Random Matrices and the Statistical Theory of Energy Levels, chapters 5 and 6. New York: Academic Press 1967 · Zbl 0925.60011
[5] Des Cloiseaux, J., Mehta, M.L.: J. Math. Phys.14, 1648–1650 (1973) · Zbl 0268.60058
[6] Gaudin, M.: Nucl. Phys.25, 447–458 (1961) · Zbl 0107.44605
[7] Dyson, F.J.: J. Math. Phys.3, 157–165 (1962)
[8] Widom, H.: Indiana Univ. Math. J.21, 277–283 (1971) · Zbl 0223.33015
[9] Grenander, U., Szegö, G.: Toeplitz Forms and their Applications, p. 76. Berkeley: University of California Press 1958 · Zbl 0080.09501
[10] Widom, H.: Am. J. Math.95, 333–383 (1973) · Zbl 0275.45006
[11] McCoy, B.M., Wu, T.T.: The Two-dimensional Ising Model. Cambridge, Mass.: Harvard University Press 1973 · Zbl 1094.82500
[12] Jost, R.: Helv. Phys. Acta20, 256–266 (1947)
[13] Levinson, N.: Kgl. Dansk. Vidensk. Selsk. Mat.-fys. Medd.25, No. 9, 1–29 (1949)
[14] Gaudin, M.: Letter to M.L. Mehta dated May 23, 1967. I am indebted to M.L. Mehta for a copy of this letter
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.