Abstract
We consider the Hamiltonian associated with a system of two particles (bosons) on a two-dimensional lattice \(\mathbb{Z}^{2}\) with a potential of a certain type. The Schrödinger operator \(H(\mathbf{k})\) of the system for \(\mathbf{k}=\boldsymbol{\pi}=(\pi,\pi)\) (where \(\mathbf{k}=(k_{1},k_{2})\) ) is the total quasimomentum) has an infinite number of eigenvalues. It is shown that \(z_{0}(\boldsymbol{\pi})=4-\bar{{v}}(0)\) is simple, \(z_{1}(\boldsymbol{\pi})=4-\bar{{v}}(1)\) is a double, \(z_{2}(\boldsymbol{\pi})=4-\bar{{v}}(2)\) is a fourfold eigenvalue, while the remaining eigenvalues \(z_{n}(\boldsymbol{\pi})=4-\bar{{v}}(n),n\geq 3,\) are fivefold. We prove that all multiple eigenvalues of the \(H(\boldsymbol{\pi})\) are split into non-degenerate eigenvalues. We obtain asymptotic formulas with the accuracy of \(\beta^{2}\) for eigenvalues of the Schrödinger operator \(H((\pi-2\beta,\pi))\) .
