On a method in scattering theory

Akchurin, È.; Minlos, R.

doi:10.1090/S0077-1554-2012-00194-0

On a method in scattering theory
HTML articles powered by AMS MathViewer

by È. R. Akchurin and R. A. Minlos
Translated by: V. E. Nazaikinskii

Trans. Moscow Math. Soc. 2011, 143-156

DOI: https://doi.org/10.1090/S0077-1554-2012-00194-0

Published electronically: January 12, 2012

PDF | Request permission

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

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, DOI 10.3103/S0027132210060069
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
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, DOI 10.1090/mmono/105
I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 1, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Properties and operations; Translated from the Russian by Eugene Saletan. MR 0435831
I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 2. Spaces of fundamental and generalized functions, Academic Press, New York-London, 1968. Translated from the Russian by Morris D. Friedman, Amiel Feinstein and Christian P. Peltzer. MR 0230128
Michael Reed and Barry Simon, Methods of modern mathematical physics. III, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. Scattering theory. MR 529429
Frédéric Riesz and Béla Sz.-Nagy, Leçons d’analyse fonctionnelle, Gauthier-Villars, Éditeur-Imprimeur-Libraire, Paris; Akadémiai Kiadó, Budapest, 1965 (French). Quatrième édition. Académie des Sciences de Hongrie. MR 0179567
A. N. Kolmogorov and S. V. Fomin, Èlementy teorii funktsiĭ i funktsional′nogo analiza, Izdat. “Nauka”, Moscow, 1968 (Russian). Second edition, revised and augmented. MR 0234241
W. V. Lovitt, Linear integral equations, Dover, New York, 1950.
N. I. Muskhelishvili, Singulyarnye integral′nye uravneniya, Third, corrected and augmented edition, Izdat. “Nauka”, Moscow, 1968 (Russian). Granichnye zadachi teorii funktsiĭ i nekotorye ikh prilozheniya k matematicheskoĭ 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
I. I. Privalov, Graničnye svoĭstva analitičeskih funkciĭ, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1950 (Russian). 2d ed.]. MR 0047765
M. V. Fedoryuk, Asimptotika: integraly i ryady, Spravochnaya Matematicheskaya Biblioteka. [Mathematical Reference Library], “Nauka”, Moscow, 1987 (Russian). MR 950167
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)
I. M. Gel′fand, D. A. Raĭkov, and G. E. Šilov, Kommutativnye normirovannye kol′tsa, Sovremennye Problemy Matematiki, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1960 (Russian). MR 0123921
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