On the number of eigenvalues of a model operator on a one-dimensional lattice

A. A. Imomov ; I. N. Bozorov ; A. M. Hurramov

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 78 (2022), pp. 22-37

Résumé

A model operator $h_{\mu}(k)$, $k\in(-\pi,\pi]$, corresponding to the Hamiltonian of a system of two arbitrary quantum particles on a one-dimensional lattice with a special dispersion function is considered. The function describes the transfer of a particle from site to sites interacting using a short-range attraction potential $\nu_{\mu}$, $\mu = (\mu_{0},\mu_{1},\mu_{2},\mu_{3}) \in\mathbb{R}_{+}^{4}$. The detailed descriptions of changes in the number of eigenvalues of the energy operator $h_{\mu}(k)$, $k\in(-\pi,\pi]$, relative to values of the particle interaction vector $\mu\in\mathbb{R}_{+}^{4}$ and the total quasi-momentum $k\in \mathbb{T}$ of the system of two particles is presented.

Détail

Keywords: Schrodinger operator, Hamiltonian of a system of two particles, one-dimensional lattice, invariant subspaces, eigenvalue, essential spectrum, unitarily equivalent operator, asymptotics for the Fredholm determinant.
Mots-clés : dispersion relations