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.


81U40 Inverse scattering problems in quantum theory
47A40 Scattering theory of linear operators
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[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
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