Geodesic
Positivity of the two-particle Hamiltonian on a lattice
Teoretičeskaâ i matematičeskaâ fizika, Tome 153 (2007) no. 3, pp. 381-387 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider a two-particle Hamiltonian on the $d$-dimensional lattice $\mathbb Z^d$. We find a sufficient condition for the positivity of a family of operators $h(k)$ appearing after the "separation of the center of mass" of a system of two particles depending on the values of the total quasimomentum $k\in T^d$ (where $T^d$ is a $d$-dimensional torus). We use the obtained result to show that the operator $h(k)$ has positive eigenvalues for nonzero $k\in T^d$.
Keywords: two-particle Hamiltonian on a lattice, virtual level, regular point, positive operator, discrete spectrum. 
@article{TMF_2007_153_3_a4,
     author = {M. I. Muminov},
     title = {Positivity of the two-particle {Hamiltonian} on a lattice},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {381--387},
     year = {2007},
     volume = {153},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2007_153_3_a4/}
}
TY  - JOUR
AU  - M. I. Muminov
TI  - Positivity of the two-particle Hamiltonian on a lattice
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2007
SP  - 381
EP  - 387
VL  - 153
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2007_153_3_a4/
LA  - ru
ID  - TMF_2007_153_3_a4
ER  - 
%0 Journal Article
%A M. I. Muminov
%T Positivity of the two-particle Hamiltonian on a lattice
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2007
%P 381-387
%V 153
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2007_153_3_a4/
%G ru
%F TMF_2007_153_3_a4
M. I. Muminov. Positivity of the two-particle Hamiltonian on a lattice. Teoretičeskaâ i matematičeskaâ fizika, Tome 153 (2007) no. 3, pp. 381-387. http://geodesic.mathdoc.fr/item/TMF_2007_153_3_a4/

[1] L. D. Faddeev, Tr. MIAN, 69, 1963, 1–122 | MR | Zbl

[2] D. C. Mattis, Rev. Mod. Phys., 58 (1986), 361–379 | DOI | MR

[3] S. Albeverio, S. N. Lakaev, K. A. Makarov, Z. I. Muminov, Commun. Math. Phys., 262 (2006), 91–115 | DOI | MR | Zbl

[4] E. L. Lakshtanov, R. A. Minlos, Funkts. analiz i ego prilozh., 38:3 (2004), 52–69 | DOI | MR | Zbl

[5] E. L. Lakshtanov, R. A. Minlos, Funkts. analiz i ego prilozh., 39:1 (2005), 39–55 | DOI | MR | Zbl

[6] P. A. Faria da Veiga, L. Ioriatti, M. O'Carroll, Phys. Rev. E, 66 (2002), 016130 | DOI | MR

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

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

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