Geodesic
Positivity of eigenvalues of the two-particle Schrödinger operator on a lattice
Teoretičeskaâ i matematičeskaâ fizika, Tome 178 (2014) no. 3, pp. 390-402 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider the family $H(k)$ of two-particle discrete Schrödinger operators depending on the quasimomentum of a two-particle system $k\in\mathbb T^d$, where $\mathbb T^d$ is a $d$-dimensional torus. This family of operators is associated with the Hamiltonian of a system of two arbitrary particles on the $d$-dimensional lattice $\mathbb Z^d$, $d\ge3$, interacting via a short-range attractive pair potential. We prove that the eigenvalues of the Schrödinger operator $H(k)$ below the essential spectrum are positive for all nonzero values of the quasimomentum $k\in\mathbb T^d$ if the operator $H(0)$ is nonnegative. We establish a similar result for the eigenvalues of the Schrödinger operator $H_+(k)$, $k\in\mathbb T^d$, corresponding to a two-particle system with repulsive interaction.
Keywords: discrete Schrödinger operator, system quasimomentum, Hamiltonian, repulsive interaction, virtual level, eigenvalue, lattice. 
@article{TMF_2014_178_3_a4,
     author = {S. N. Lakaev and Sh. U. Alladustov},
     title = {Positivity of eigenvalues of the~two-particle {Schr\"odinger} operator on a~lattice},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {390--402},
     year = {2014},
     volume = {178},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2014_178_3_a4/}
}
TY  - JOUR
AU  - S. N. Lakaev
AU  - Sh. U. Alladustov
TI  - Positivity of eigenvalues of the two-particle Schrödinger operator on a lattice
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2014
SP  - 390
EP  - 402
VL  - 178
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2014_178_3_a4/
LA  - ru
ID  - TMF_2014_178_3_a4
ER  - 
%0 Journal Article
%A S. N. Lakaev
%A Sh. U. Alladustov
%T Positivity of eigenvalues of the two-particle Schrödinger operator on a lattice
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2014
%P 390-402
%V 178
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2014_178_3_a4/
%G ru
%F TMF_2014_178_3_a4
S. N. Lakaev; Sh. U. Alladustov. Positivity of eigenvalues of the two-particle Schrödinger operator on a lattice. Teoretičeskaâ i matematičeskaâ fizika, Tome 178 (2014) no. 3, pp. 390-402. http://geodesic.mathdoc.fr/item/TMF_2014_178_3_a4/

[1] S. N. Lakaev, S. S. Ulashov, TMF, 170:3 (2012), 393–408 | DOI | DOI | MR | Zbl

[2] K. Winker, G. Thalhammer, F. Lang, R. Grimm, J. Hesker Denschlog, A. J. Daley, A. Kantian, H. P. Büchler, P. Zoller, Nature, 441:7095 (2006), 853–856, arXiv: cond-mat/0605196 | DOI

[3] S. Albeverio, S. N. Lakaev, Z. I. Muminov, Ann. H. Poincaré, 5:4 (2004), 743–772 | DOI | MR | Zbl

[4] S. Albeverio, G. F. Dell Antonio, S. N. Lakaev, J. Phys. A, 40:49 (2007), 14819–14842, arXiv: math/0703191 | DOI | MR | Zbl

[5] M. E. Muminov, TMF, 153:3 (2007), 381–387 | DOI | DOI | MR | Zbl

[6] S. Albeverio, S. N. Lakaev, A. M. Khalkhuzhaev, Markov Proc. Relat. Fields, 18:3 (2012), 387–420 | MR | Zbl

[7] S. N. Lakaev, Funk. analiz i ego ppil., 27:3 (1993), 15–28 | DOI | MR | Zbl

[8] S. Albeverio, S. N. Lakaev, K. A. Makarov, Z. I. Muminov, Commun. Math. Phys., 262:1 (2006), 91–115, arXiv: math-ph/0501013 | DOI | MR | Zbl

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

[10] A. V. Sobolev, Commun. Math. Phys., 156:1 (1993), 127–168 | DOI | MR

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

[12] D. R. Yafaev, Zap. nauchn. sem. LOMI, 51 (1975), 203–216 ; D. R. Yafaev, Scattering Theory: Some Old and New Problems, Lecture Notes in Mathematics, 1735, Springer, Berlin, 2000 | DOI | MR | Zbl | DOI | MR | Zbl

[13] V. Bach, W. de Siqueira Pedra, S. Lakaev, Bounds on the discrete spectrum of lattice Schrödinger operators, Preprint mp-arc 10-143, University of Texas, Austin, 2011 | Zbl