Number of Bound States of the Hamiltonian of a Lattice Two-boson System with Interactions up to the Next Neighboring Sites

• Published in: Lobachevskii journal of mathematics Vol. 45; no. 12; pp. 6526 - 6537
• Main Authors: Lakaev, S. N., Khamidov, Sh. I., Akhmadova, M. O.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.12.2024; Springer Nature B.V
• Subjects: Algebra; Analysis; Bosons; Eigenvalues; Geometry; Lower bounds; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Operators (mathematics); Probability Theory and Stochastic Processes; Schrodinger equation; Subspaces; essential spectrum; bound states; two-particle system; lattice Schrödinger operator; Fredholm determinant; 2020 Mathematics Subject Classification: Primary 81Q10, Secondary: 47A10, 47A55, 47A75, 47J10, 34L40
• ISSN: 1995-0802; 1818-9962
• DOI: 10.1134/S1995080224607185

We study the family , of discrete Schrödinger operators, associated to the Hamiltonian of a system of two identical bosons on the two-dimensional lattice interacting through on one site, nearest-neighbor sites and next-nearest-neighbor sites with interaction magnitudes and respectively. We prove there existence an important invariant subspace of operator such that the restriction of the operator on this subspace has at most two eigenvalues lying both as below the essential spectrum as well as above it, depending on the interaction magnitude (only). We also give a sharp lower bound for the number of eigenvalues of .