Geodesic
Formula for the number of eigenvalues of a three-particle Schrödinger operator on a lattice
Teoretičeskaâ i matematičeskaâ fizika, Tome 164 (2010) no. 1, pp. 46-61 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider a system of three arbitrary quantum particles on a three-dimensional lattice that interact via short-range attractive potentials. We obtain a formula for the number of eigenvalues in an arbitrary interval outside the essential spectrum of the three-particle discrete Schrödinger operator and find a sufficient condition for the discrete spectrum to be finite. We give an example of an application of our results.
Keywords: discrete spectrum, essential spectrum, Schrödinger operator, positive operator, compact operator. 
@article{TMF_2010_164_1_a2,
     author = {M. I. Muminov},
     title = {Formula for the~number of eigenvalues of a~three-particle {Schr\"odinger} operator on a~lattice},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {46--61},
     year = {2010},
     volume = {164},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2010_164_1_a2/}
}
TY  - JOUR
AU  - M. I. Muminov
TI  - Formula for the number of eigenvalues of a three-particle Schrödinger operator on a lattice
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2010
SP  - 46
EP  - 61
VL  - 164
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2010_164_1_a2/
LA  - ru
ID  - TMF_2010_164_1_a2
ER  - 
%0 Journal Article
%A M. I. Muminov
%T Formula for the number of eigenvalues of a three-particle Schrödinger operator on a lattice
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2010
%P 46-61
%V 164
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2010_164_1_a2/
%G ru
%F TMF_2010_164_1_a2
M. I. Muminov. Formula for the number of eigenvalues of a three-particle Schrödinger operator on a lattice. Teoretičeskaâ i matematičeskaâ fizika, Tome 164 (2010) no. 1, pp. 46-61. http://geodesic.mathdoc.fr/item/TMF_2010_164_1_a2/

[1] A. V. Sobolev, Comm. Math. Phys., 156:1 (1993), 101–126 | DOI | MR | Zbl

[2] H. Tamura, Nagoya Math. J., 130 (1993), 55–83 | DOI | MR | Zbl

[3] S. N. Lakaev, Z. E. Muminov, Funkts. analiz i ego pril., 37:3 (2003), 80–84 | DOI | MR | Zbl

[4] Zh. I. Abdullaev, S. N. Lakaev, TMF, 136:2 (2003), 231–245 | DOI | MR | Zbl

[5] S. Albeverio, S. N. Lakaev, Z. I. Muminov, Math. Nachr., 280:7 (2007), 699–716 ; arXiv: math-ph/0312050 | DOI | MR | Zbl

[6] M. E. Muminov, TMF, 159:2 (2009), 299–317 | DOI | MR | Zbl

[7] S. N. Lakaev, M. E. Muminov, TMF, 135:3 (2003), 478–503 | DOI | MR | Zbl

[8] A. A. Vladimirov, Matem. zametki, 74:6 (2003), 838–847 | DOI | MR | Zbl

[9] M. E. Muminov, Matem. zametki, 82:1 (2007), 75–83 | DOI | MR | Zbl

[10] P. Khalmosh, Gilbertovo prostranstvo v zadachakh, Mir, M., 1970 | MR | MR | Zbl

[11] L. D. Faddeev, Matematicheskie voprosy kvantovoi teorii rasseyaniya dlya sistemy trekh chastits, Tr. MIAN SSSR, 69, ed. I. G. Petrovskii, Izd-vo AN SSSR, M.–L., 1963 | MR | Zbl

[12] M. Rid, B. Saimon, Metody sovremennoi matematicheskoi fiziki, v. 4, Analiz operatorov, Mir, M., 1982 | MR | MR | Zbl

[13] M. Sh. Birman, M. Z. Solomyak, Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve, Izd-vo LGU, L., 1980 | MR | MR | Zbl