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Ahiezer-Kac type Fredholm determinant asymptotics for convolution operators with rational symbols

Authors: Sergio Albeverio and Konstantin A. Makarov
Journal: Trans. Amer. Math. Soc. 353 (2001), 1985-1993
MSC (1991): Primary 45P05, 47B35; Secondary 47A68, 47G10
Published electronically: January 10, 2001
MathSciNet review: 1813603
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Fredholm determinant asymptotics of convolution operators on large finite intervals with rational symbols having real zeros are studied. The explicit asymptotic formulae obtained can be considered as a direct extension of the Ahiezer-Kac formula to symbols with real zeros.

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  • 1. N. I. Ahiezer, The continuous analogue of some theorems on Toeplitz matrices, Ukrain. Mat. Zh. 16 (1964) 455-462; English transl., Amer. Math. Soc. Transl. (2) 50 (1966), 295-316. MR 30:411
  • 2. S. Albeverio, R. Høegh-Krohn, and T. T. Wu, A class of exactly solvable three-body quantum mechanical problems and universal low energy behavior, Phys. Lett. A 83 (1981), 105-109. MR 82h:81017
  • 3. S. Albeverio, S. Lakaev, and K. A. Makarov, The Efimov effect and an extended Szegö-Kac limit theorem, Lett. Math. Phys. 43 (1998), 73-85. MR 99e:47031
  • 4. S. Albeverio and K. A. Makarov, Non trivial attractors in a model connected with the three-body Quantum Problem, Acta Applicandae Math. 48 (1997), 113-184. MR 99a:47033
  • 5. S. Albeverio and K. A. Makarov, Limit behavior in a singular perturbation problem, regularized convolution operators and the three-body problem, Differential and Integral Operators (Regensburg, 1995), Operator Theory: Advances and Applications, vol. 102, Birkhäuser, Basel, 1998, pp. 1-10. MR 99i:81261
  • 6. E. Basor, Asymptotic formulas for Toeplitz determinants, Trans. Amer. Math. Soc. 239 (1978), 33-65. MR 58:12484
  • 7. A. Böttcher, Wiener-Hopf determinants with rational symbols, Math. Nachr. 144 (1989), 39-64. MR 91g:47017
  • 8. A. Böttcher and B. Silbermann, The asymptotic behavior of Toeplitz determinants for generating functions with zeros of integer orders, Math. Nachr. 102 (1981), 79-105. MR 83f:47022
  • 9. A. Böttcher and B. Silbermann, Wiener-Hopf determinants with symbols having zeros of analytic type, Seminar Analysis 1982/1983, 224-243, Akad. Wiss. DDR, Inst. Math., Berlin, 1983. MR 85e:47035
  • 10. A. Böttcher and B. Silbermann, Analysis of Toeplitz Operators, Springer-Verlag, Berlin, 1990. MR 92e:47001
  • 11. K. M. Day, Toeplitz matrices generated by the Laurent series expansion of an arbitrary rational function, Trans. Amer. Math. Soc. 206 (1975), 224-245. MR 52:708
  • 12. M. E. Fisher and R. E. Hartwig, Toeplitz determinants: some applications, theorems, and conjectures. Adv. Chem. Phys. 15 (1968), 333-353.
  • 13. M. E. Fisher and R. E. Hartwig, Asymptotic behavior of Toeplitz matrices and determinants, Arch. Rat. Mech. Anal. 32 (1969), 190-225. MR 38i:4888
  • 14. L. D. Faddeev and S. P. Merkuriev, Quantum scattering theory for several particle systems. Kluwer, Dordrecht, 1993. MR 94j:81276
  • 15. I. Gohberg, S. Goldberg, and M. A. Kaashoek, Classes of linear operators, Vol I. Operator Theory: Advances and Applications 49, Birkhäuser, 1990. MR 93d:47002
  • 16. I. Gohberg and M. G. Krein, Introduction to the theory of linear nonselfadjoint operators in Hilbert space. Nauka, Moscow 1965 (Russian), Engl. trans.: Amer. Math. Soc. Transl. of Math. Monographs 18, Providence, R. I. 1969. MR 36:3137; MR 39:7447
  • 17. I. Gohberg, M. A. Kaashoek, and F. van Schagen, Szego-Kac-Achiezer formulas in terms of realizations of the symbols, J. Func. Anal. 74 (1987), 24-51. MR 88m:47043
  • 18. M. Kac, Toeplitz matrices, translation kernels and a related problem in probability theory, Duke Math. J. 21 (1954), 501-509. MR 16:31a
  • 19. M. Kac, W. L. Murdock, and G. Szegö, On the eigenvalues of certain Hermitian forms, J. Rational Mechanics and Analysis 2 (1953), 767-800. MR 15:538b
  • 20. H. Widom, Toeplitz determinants with singular generating functions, Amer. J. Math. 95 (1973), 333-383. MR 48:9441

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Additional Information

Sergio Albeverio
Affiliation: Institut für Angewandte Mathematik, Universität Bonn, 53115 Bonn, Germany, SFB 237 (Essen-Bochum-Düsseldorf), BiBoS (Bielefeld-Bochum/Bonn), CERFIM (Locarno)

Konstantin A. Makarov
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211

Received by editor(s): October 17, 1997
Published electronically: January 10, 2001
Article copyright: © Copyright 2001 by the authors

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