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Transactions of the Moscow Mathematical Society

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On a method in scattering theory

Authors: È. R. Akchurin and R. A. Minlos
Translated by: V. E. Nazaikinskii
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 72 (2011), vypusk 2.
Journal: Trans. Moscow Math. Soc. 2011, 143-156
MSC (2010): Primary 47A40; Secondary 35P25, 35Q40
Published electronically: January 12, 2012
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Abstract: We use the well-studied Friedrichs model to showcase a new method for proving the asymptotic completeness of two operators, which in our case are the Friedrichs operator $ A$ and the operator obtained from $ A$ by omitting the integral term. Technically, the problem is reduced to a detailed analysis of the Fredholm determinant and minor of an auxiliary integral operator.

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  • 1. È. R. Akchurin and R. A. Minlos, A new method in the scattering theory, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 65 (2010), no. 6, 27-32; English transl., Moscow Univ. Math. Bull. 65 (2010), no. 6, 247-251. MR 2814987
  • 2. L. D. Faddeev, On the Friedrichs model in the theory of perturbations of a continuous spectrum, Trudy Mat. Inst. Steklov 73 (1964), 292-313; English transl., Amer. Math. Soc., Transl. Ser. 2 , vol. 62, Amer. Math. Soc., Providence, RI, 1967, pp. 177-203. MR 0178362 (31:2620)
  • 3. D. R. Yafaev, Mathematical scattering theory: General theory, Izdat. Sankt-Peterburg. Univ., St. Petersburg, 1994; English transl., Transl. of Math. Monographs, vol. 105, Amer. Math. Soc., Providence, RI, 1992 (reprinted with corrections in 1998). MR 1180965 (94f:47012)
  • 4. I. M. Gelfand and G. E. Shilov, Generalized functions. Vol. 1. Properties and operations, Gostekhizdat, Moscow, 1959; English transl., Academic Press, New York-London, 1964. MR 0435831 (55:8786a)
  • 5. I. M. Gelfand and G. E. Shilov, Generalized functions. Vol. 2. Spaces of fundamental and generalized functions, Gostekhizdat, Moscow, 1958; English transl., Academic Press, New York-London, 1968. MR 0230128 (37:5693)
  • 6. M. Reed and B. Simon, Methods of modern mathematical physics. III. Scattering theory, Academic Press, New York-London, 1979. MR 529429 (80m:81085)
  • 7. F. Riesz and B. Sz.-Nagy, Leçons d'analyse fonctionnelle, Gauthier-Villars, Paris; copublished with Akadémiai Kiadó, Budapest, 1965. (French) MR 0179567 (31:3815)
  • 8. A. N. Kolmogorov and S. V. Fomin, Elements of the theory of functions and functional analysis, 3rd ed., Nauka, Moscow, 1972; English transl. of the 2nd ed. (1968), Prentice-Hall, Englewood Cliffs, NJ, 1970. MR 0234241 (38:2559)
  • 9. W. V. Lovitt, Linear integral equations, Dover, New York, 1950.
  • 10. N. I. Muskhelishvili, Singular integral equations, Nauka, Moscow, 1968; English transl. of the 2nd ed. (1962), Wolters-Noordhoff, Groningen, 1967. MR 0355495 (50:7969)
  • 11. I. I. Privalov, Boundary properties of analytic functions, 2nd ed., Gosudarstv. Izdat. Tekhn.-Teor. Lit., Moscow, 1950. (Russian) MR 0047765 (13:926h)
  • 12. M. V. Fedoryuk, Asymptotics. Integrals and series, Nauka, Moscow, 1987. MR 950167 (89j:41045)
  • 13. È. R. Akchurin and R. A. Minlos, Scattering theory for a class of two-particle operators of mathematical physics (the case of weak interaction), Izv. Ross. Akad. Nauk Ser. Mat. (to appear)
  • 14. I. M. Gelfand, D. A. Raikov, and G. E. Shilov, Commutative normed rings, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1960; English transl., Chelsea Publishing Co., New York, 1964. MR 0123921 (23:A1242)
  • 15. V. I. Paraska, On asymptotics of eigenvalues and singular numbers of linear operators which increase smoothness, Mat. Sb. 68 (110) (1965), no. 4, 623-631. (Russian) MR 0199749 (33:7892)

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Additional Information

È. R. Akchurin
Affiliation: Mechanics and Mathematics Faculty, Moscow State University, Moscow 119991, Russian Federation

R. A. Minlos
Affiliation: Institute for Information Transmission Problems, Moscow 127994, Russian Federation

Keywords: Asymptotic completeness, Friedrichs model, wave operators, Fredholm minor, Fredholm determinant, stationary phase method.
Published electronically: January 12, 2012
Article copyright: © Copyright 2011 American Mathematical Society

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