Speaker
Description
We consider the lattice Schr\"odinger operator $H_{\gamma \lambda \mu}(K)$ associated with a system of two identical particles on the two-dimensional square lattice $\mathbb{Z}^2$. It is assumed that the center-of-mass quasimomentum $K$ equals zero and that the particles may interact with each other either on-site or on the first and second nearest neighbouring sites in the lattice. These interactions have magnitudes $\gamma$, $\lambda$ and $\mu$, respectively. We study the discrete spectrum of parts of the operator $H_{\gamma \lambda \mu}(0)$ in its certain reducing subspaces (the fermion subspace and a part of the boson subspace). We partition the corresponding $(\lambda,\mu)$- and $(\gamma,\lambda,\mu)$-parameter sets into connected components such that, in each component, the involved part of the Hamiltonian $H_{\gamma \lambda \mu}(0)$ has fixed numbers of eigenvalues below the bottom of the essential spectrum and above its top.
The talk is based on joint works [1] with S.N.Lakaev, S.Kh.,Abdukhakimov and [2] with S.N.Lakaev, M.O.Akhmadova.
This research was supported in part by the Ministry of Innovative Development of the Republic of Uzbekistan (Grant No. FZ-20200929224).
[1] S.N. Lakaev, A.K. Motovilov, and S.Kh.Abdukhakimov, "Two-fermion lattice Hamiltonian with first and second nearest-neighboring-site interactions", J. Phys. A: Math. Theor. 56 (2023), 315202 [23 pages].
[2] S.N. Lakaev, A.K. Motovilov, and M.O.Akhmadova, "A two-boson lattice Hamiltonian with interactions up to next-neighboring sites", arXiv:2410.07070 (2024).
