The Infiniteness of the Number of Eigenvalues of the Schrödinger Operator of a System of Two Particles on a Lattice
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 $$H((\pi-2\beta,\pi))$$ .