Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Transactions of the Moscow Mathematical Society
Transactions of the Moscow Mathematical Society
ISSN 1547-738X(online) ISSN 0077-1554(print)

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
Posted: January 12, 2012
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. È. R. Akchurin and R. A. Minlos, On a new method in scattering theory, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 6 (2010), 27–32 (Russian, with English and Russian summaries); English transl., Moscow Univ. Math. Bull. 65 (2010), no. 6, 247–251. MR 2814987 (2012i:47015), http://dx.doi.org/10.3103/S0027132210060069
  • 2. L. D. Faddeev, On a model of Friedrichs in the theory of perturbations of the continuous spectrum, Trudy Mat. Inst. Steklov 73 (1964), 292–313 (Russian). MR 0178362 (31 #2620)
  • 3. D. R. Yafaev, Mathematical scattering theory, Translations of Mathematical Monographs, vol. 105, American Mathematical Society, Providence, RI, 1992. General theory; Translated from the Russian by J. R. Schulenberger. MR 1180965 (94f:47012)
  • 4. I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 1, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1964 [1977]. Properties and operations; Translated from the Russian by Eugene Saletan. MR 0435831 (55 #8786a)
  • 5. I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 2. Spaces of fundamental and generalized functions, Translated from the Russian by Morris D. Friedman, Amiel Feinstein and Christian P. Peltzer, Academic Press, New York, 1968. MR 0230128 (37 #5693)
  • 6. Michael Reed and Barry Simon, Methods of modern mathematical physics. III, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1979. Scattering theory. MR 529429 (80m:81085)
  • 7. Frédéric Riesz and Béla Sz.-Nagy, Leçons d’analyse fonctionnelle, Gauthier-Villars, Editeur, 1965 (French). Quatrième édition. Académie des Sciences de Hongrie. MR 0179567 (31 #3815)
  • 8. A. N. Kolmogorov and S. V. Fomin, Elementy teorii funktsii i funktsionalnogo analiza, Second edition, revised and augmented, Izdat. “Nauka”, Moscow, 1968 (Russian). MR 0234241 (38 #2559)
  • 9. W. V. Lovitt, Linear integral equations, Dover, New York, 1950.
  • 10. N. I. \cyr{M}uskhelishvili, Singulyarnye integralnye uravneniya, Third, corrected and augmented edition, Izdat. “Nauka”, Moscow, 1968 (Russian). Granichnye zadachi teorii funktsii i nekotorye ikh prilozheniya k matematicheskoi fizike. [Boundary value problems in the theory of function and some applications of them to mathematical physics]; With an appendix by B. Bojarski. MR 0355495 (50 #7969)
  • 11. I. I. Privalov, Graničnye svoĭstva analitičeskih funkciĭ, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1950 (Russian). 2d ed.]. MR 0047765 (13,926h)
  • 12. M. V. Fedoryuk, Asimptotika: integraly i ryady, \cyr Spravochnaya Matematicheskaya Biblioteka. [Mathematical Reference Library], “Nauka”, Moscow, 1987 (Russian). 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. Gel′fand, D. A. Raĭkov, and G. E. Šilov, Kommutativnye normirovannye koltsa, Sovremennye Problemy Matematiki, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1960 (Russian). MR 0123921 (23 #A1242)
  • 15. V. I. Paraska, On asymptotics of eigenvalues and singular numbers of linear operators which increase smoothness, Mat. Sb. (N.S.) 68 (110) (1965), 623–631 (Russian). MR 0199749 (33 #7892)

Similar Articles

Retrieve articles in Transactions of the Moscow Mathematical Society with MSC (2010): 47A40, 35P25, 35Q40

Retrieve articles in all journals with MSC (2010): 47A40, 35P25, 35Q40


Additional Information

È. R. Akchurin
Affiliation: Mechanics and Mathematics Faculty, Moscow State University, Moscow 119991, Russian Federation
Email: eakchurin@gmail.com

R. A. Minlos
Affiliation: Institute for Information Transmission Problems, Moscow 127994, Russian Federation
Email: minl@iitp.ru

DOI: http://dx.doi.org/10.1090/S0077-1554-2012-00194-0
PII: S 0077-1554(2012)00194-0
Keywords: Asymptotic completeness, Friedrichs model, wave operators, Fredholm minor, Fredholm determinant, stationary phase method.
Posted: January 12, 2012
Article copyright: © Copyright 2011 American Mathematical Society