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Nanosystems: Phys. Chem. Math., 2022, 13 (3), 237ā€“244

Conditions for the existence of bound states of a two-particle Hamiltonian on a three-dimensional lattice

M. I. Muminov – Samarkand State University, Samarkand, 140104; V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of sciences 100174, Tashkent, Uzbekistan; mmuminov@mail.ru
A. M. Khurramov – Samarkand State University, Samarkand, 140104, Uzbekistan; xurramov@mail.ru
I. N. Bozorov – Samarkand State University, Samarkand, 140104, Uzbekistan; islomnb@mail.ru

Corresponding author: M.I. Muminov, mmuminov@mail.ru

DOI 10.17586/2220-8054-2022-13-3-237-244

ABSTRACT The Hamiltonian h of the system of two quantum particles moving on a 3-dimensional lattice interacting via some attractive potential is considered. Conditions for the existence of eigenvalues of the two-particle Schrödinger operator hμ(k), k∈T3, μ∈R associated to the Hamiltonian h, are studied depending on the energy of the particle interaction μ∈R and total quasi-momentum k∈T3 (T3 ā€“ three-dimensional torus).

KEYWORDS two-particle Hamiltonian, invariant subspace, unitary equivalent operator, virtual level, multiplicity of virtual level, eigenvalue.

ACKNOWLEDGEMENTS The authors express their kind gratitude to the referee for valuable remarks. The work was supported by the Fundamental Science Foundation of Uzbekistan, Grant No. FZā€“20200929224.

FOR CITATION Muminov M.I., Khurramov A.M., Bozorov I.N. Conditions for the existence of bound states of a two-particle Hamiltonian on a three-dimensional lattice. Nanosystems: Phys. Chem. Math., 2022, 13 (3), 237ā€“244.

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