Geodesic
Spectral properties of a~two-particle Hamiltonian on a~lattice
Teoretičeskaâ i matematičeskaâ fizika, Tome 177 (2013) no. 3, pp. 482-496

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We consider a system of two arbitrary quantum particles on a three-dimensional lattice with some dispersion functions (describing particle transport from a site to a neighboring site). The particles interact via an attractive potential at only the nearest-neighbor sites. We study how the number of eigenvalues of a family of operators $h(k)$ depends on the particle interaction energy and the total quasimomentum $k\in\mathbb T^3$, where $\mathbb T^3$ is a three-dimensional torus. We find the conditions under which the operator $h(\mathbf 0)$ has a double or triple virtual level at zero depending on the particle interaction energy.
Keywords: two-particle Hamiltonian on a lattice, virtual level, virtual-level multiplicity, eigenvalue, positive operator. 
@article{TMF_2013_177_3_a4,
     author = {M. I. Muminov and A. M. Hurramov},
     title = {Spectral properties of a~two-particle {Hamiltonian} on a~lattice},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {482--496},
     publisher = {mathdoc},
     volume = {177},
     number = {3},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2013_177_3_a4/}
}
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M. I. Muminov; A. M. Hurramov. Spectral properties of a~two-particle Hamiltonian on a~lattice. Teoretičeskaâ i matematičeskaâ fizika, Tome 177 (2013) no. 3, pp. 482-496. http://geodesic.mathdoc.fr/item/TMF_2013_177_3_a4/