Preview

Nanosystems: Physics, Chemistry, Mathematics

Advanced search

Efimov’s effect for partial integral operators of Fredholm type

Abstract

We study the existence of an infinite number of eigenvalues (the existence of Efimov’s effect) for a self-adjoint partial integral operators. We prove a theorem on the necessary and sufficient conditions for the existence of Efimov’s effect for the Fredholm type partial integral operators.

About the Authors

Yu. Kh. Eshkabilov
Department of Mechanics and Mathematics, National University of Uzbekistan
Uzbekistan

 Tashkent



R. R. Kucharov
Department of Mechanics and Mathematics, National University of Uzbekistan
Uzbekistan

 Tashkent



References

1. Kalitvin A.S.On partial integral operators incontact problems of elasticity. (in Russian) Proc. 26 Voronezh Winter School, 54, (1994).

2. Kovalenko E.V. On the approximate solution of one type of integral equations a rising in elasticity type mathematical physics(in Russian). Izv. Akad. Nauk Arm. SSR,34, No 5,P.14–26 (1981).

3. Vekua I.N. New Methods of Solving Elliptic Equations (in Russian).Gostekhizdat, Moscow-Leningrad (1948).

4. Aleksandrov V.M., Kovalenko E.V.On some class of integral equations arising inmixed boundary value problems of continuum mechanics. Sov. Phys. Dok l.25, No 2, P.354–356 (1980).

5. Aleksandrov V.M.,Kovalenko E.V.On the contact interaction of bodies with coatings and abrasion. Sov. Phys. Dok l.29, No 4, P.340–342 (1984).

6. Aleksandrov V.M.,Kovalenko E.V.Problems of Continuum Mechanics with Mixed Boundary Conditions(in Russian).Nauka, Moscow (1986).

7. Kalitvin A.S. On some class of partial integrale quations in aerodynamics, (in Russian). Sost. Persp. Razv. Nauch.–Tekhn. Pod. Lipetsk. Obl(Lipetsk). P.210–212 (1994).

8. Goursat E.Cours d’Analyse Mathematique. Gautheir-Villars, Paris (1943).

9. Muntz C.H.Zum dynamischen Warmeleitungs problem. Math.Z. 38, P.323–337 (1934).

10. Eshkabilov Yu.Kh. On a discrete “three-particle” Schrodinger operator in the Hubbard model. Theor. Math. Phys., 149 (2), P. 1497–1511 (2006).

11. Albeverio S., Lakaev S.N., Muminov Z. I. On the number of eigenvalues of a model operator associated to a system of three-particles on lattices Russ. J. Math. Phys. 14 (4), P. 377-387 (2007).

12. Rasulov T.Kh. Asymptotics of the discrete spectrum of a model operator assotiated with a system of three particles on a lattice. Theor. Math.Phys. 163 (1), P.429–437 (2010).

13. Eshkabilov Yu.Kh., Kucharov R.R. Essential and discrete spectra of the three-particle Schr¨ odinger operator on a lattice. Theor. Math. Phys., 170 (3), P.341-353 (2012).

14. Eshkabilov Yu.Kh. Efimov’s effect for a 3-particle model discrete Schr¨ odinger operator. Theor. Math. Phys., 164 (1), P. 896-904 (2010).

15. Appell J., Frolova E.V., Kalitvin A.S., Zabrejko P.P. Partial integral operators on C([ab] [cd]). Integral Equ. Oper. Theory, 27, P. 125–140 (1997).

16. Likhtarnikov L.M., Vitova L.Z. On the spectrum of an integral operator with partial integrals (in Russian). Litov. Mat. Sbornik, 15, No 2, P.41–47 (1975).

17. Kalitvin A.S. On the spectrum of a linear operators with partial integrals and positive kernels (in Russian). In Pribl. Funk. Spektr. Theor., Leningrad, P. 43–50 (1988).

18. Kalitvin A.S., Zabrejko P.P. On the theory of partial integral operators. J. Integral Eq. Appl., 3 (3), P. 351–382 (1991).

19. Eshkabilov Yu.Kh. Spectra of partial integral operators with a kernel of three variables. Central Eur. J. Math., 6 (1), P. 149–157 (2008).

20. Appell J., Kalitvin A.S., Zabrejko P.P. Partial Integral operators and Integro-differential Equations. (Pure and Applied Mathematics: A Series of Monographs and Textbooks, 230), Marcel Dekker, Inc., New York (2000).

21. Reed M., Simon B. Methods of Modern Mathematical Physics, Vol.1, Functional Analysis. Acad. Press, New York (1972).

22. Eshkabilov Yu.Kh. Essential and discrete spectre of partially integral operators. Siberian Advances in Mathematics, 19 (4), P. 233–244 (2009).

23. Eshkabilov Yu.Kh. On infinity of the discrete spectrum of operators in the Friedrichs model. Siberian Advances in Mathematics, 22 (1), P. 1-12 (2012).

24. Eshkabilov Yu.Kh. Perturbation spectra of multiplication with a partial integral operator (in Russian). Acta NUUz, No 2, P. 17–21 (2006).

25. Eshkabilov Yu.Kh. On the spectra of the tensor sum of compact operators (in Russian). Uzbek. Mat. J., No 3, P. 104–112 (2005).

26. Eshkabilov Yu.Kh. On infinite number of negative eigenvalues of the Friedrichs model (in Russian). Nanosystems: Physics, Chemistry, Mathematics, 3 (6), P. 16–24 (2012)


Review

For citations:


Eshkabilov Yu.Kh., Kucharov R.R. Efimov’s effect for partial integral operators of Fredholm type. Nanosystems: Physics, Chemistry, Mathematics. 2013;4(4):529-537.

Views: 2


Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.


ISSN 2220-8054 (Print)
ISSN 2305-7971 (Online)