Geodesic
On the essential spectrum of a~model operator associated with the system of three particles on a~lattice
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 3 (2011), pp. 42-51.

Voir la notice de l'article provenant de la source Math-Net.Ru

A model operator $H$ associated with the system of three-identical particles on a lattice $\mathbb{Z}^3$ is considered. The location of the essential spectrum of $H$ is described by the spectrum of the corresponding Friedrichs model, that is, the two-particle and three-particle branches of the essential spectrum of $H$ are singled out. It is proved that the essential spectrum of $H$ consists of no more than three bounded closed intervals. An appearance of two-particle branches on the both sides of the three-particle branch is shown. Moreover, we obtain an analogue of the Faddeev equation and its symmetric version, for the eigenfunctions of $H.$
Keywords: model operator, Friedrichs model, Hilbert–Schmidt class, Faddeev equation, essential spectrum. 
@article{VSGTU_2011_3_a3,
     author = {T. Kh. Rasulov},
     title = {On the essential spectrum of a~model operator associated with the system of three particles on a~lattice},
     journal = {Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences},
     pages = {42--51},
     publisher = {mathdoc},
     number = {3},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGTU_2011_3_a3/}
}
TY  - JOUR
AU  - T. Kh. Rasulov
TI  - On the essential spectrum of a~model operator associated with the system of three particles on a~lattice
JO  - Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
PY  - 2011
SP  - 42
EP  - 51
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/VSGTU_2011_3_a3/
LA  - ru
ID  - VSGTU_2011_3_a3
ER  - 
%0 Journal Article
%A T. Kh. Rasulov
%T On the essential spectrum of a~model operator associated with the system of three particles on a~lattice
%J Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
%D 2011
%P 42-51
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/VSGTU_2011_3_a3/
%G ru
%F VSGTU_2011_3_a3
T. Kh. Rasulov. On the essential spectrum of a~model operator associated with the system of three particles on a~lattice. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 3 (2011), pp. 42-51. http://geodesic.mathdoc.fr/item/VSGTU_2011_3_a3/

[1] Rid M., Saymon B., Methods of modern mathematical physics, v. IV, Analysis of operators, Mir, Moscow, 1982, 430 pp. | MR

[2] Zhislin G. M., “Discussion of the spectrum of the Schrödinger operator for systems of many particles”, Trudy Moskov. Mat. Obshch., 9 (1960), 81–120 | Zbl

[3] Hunziker W., “On the spectra of Schrödinger multi-particle Hamiltonians”, Helv. Phys. Acta, 39 (1966), 451–462 | MR | Zbl

[4] Albeverio S., Lakaev S. N., Muminov Z. I., “Schrödinger operators on lattices. The Efimov effect and discrete spectrum asymptotics”, Ann. Inst. Henri Poincare, 5:4 (2004), 743–772 | DOI | MR | Zbl

[5] Albeverio S., Lakaev S. N., Muminov Z. I., “On the structure of the essential spectrum for the three-particle Schrödinger operators on lattices”, Math. Nachr., 280:7 (2007), 699–716 | DOI | MR | Zbl

[6] 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., 1:4 (2007), 377–387 | DOI | MR

[7] Albeverio S., Lakaev S. N., Djumanova R. Kh., “The essential and discrete spectrum of a model operator associated to a system of three identical quantum particles”, Rep. Math. Phys., 63:3 (2009), 359–380 | DOI | MR | Zbl

[8] Rasulov T. Kh., “Asymptotics of the discrete spectrum of a model operator associated with a system of three particles on a lattice”, Theoret. and Math. Phys., 163:1 (2010), 429–437 | DOI | DOI | MR | Zbl

[9] Rasulov T. Kh., “The Faddeev equation and the location of the essential spectrum of a model multi-particle operator”, Russian Math. (Iz. VUZ), 52:12 (2008), 50–59 | DOI | MR | Zbl

[10] Eshkabilov Yu. Kh., “A discrete “three-particle” Schrödinger operator in the Hubbard model”, Theoret. and Math. Phys., 149:2 (2006), 1497–1511 | DOI | DOI | MR | Zbl