Abstract:
We consider the family H(k) of two-particle discrete Schrödinger operators depending on the quasimomentum of a two-particle system k∈Td, where Td 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 Zd, d⩾3, 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∈Td if the operator H(0) is nonnegative. We establish a similar result for the eigenvalues of the Schrödinger operator H+(k), k∈Td, corresponding to a two-particle system with repulsive interaction.
Citation:
S. N. Lakaev, Sh. U. Alladustov, “Positivity of eigenvalues of the two-particle Schrödinger operator on a lattice”, TMF, 178:3 (2014), 390–402; Theoret. and Math. Phys., 178:3 (2014), 336–346
This publication is cited in the following 4 articles:
S. N. Lakaev, S. Kh. Abdukhakimov, “Threshold effects in a two-fermion system on an optical lattice”, Theoret. and Math. Phys., 203:2 (2020), 648–663
S. N. Lakaev, A. T. Boltaev, “Threshold phenomena in the spectrum of the two-particle Schrödinger operator on a lattice”, Theoret. and Math. Phys., 198:3 (2019), 363–375
Sh. Yu. Kholmatov, Z. I. Muminov, “Existence of bound states of N-body problem in an optical lattice”, J. Phys. A-Math. Theor., 51:26 (2018), 265202
I. P. Popov, “Gruppovaya skorost volnovogo paketa, obrazovannogo dvumya svobodnymi identichnymi chastitsami s raznymi nerelyativistskimi skorostyami”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2015, no. 3(35), 69–72