Bound states of a two-boson system on a two-dimensional lattice

Abstract

We consider a Hamiltonian of a two-boson system on a two-dimensional lattice Z². The Schrödinger operator H(k₁, k₂) of the system for k₁ = k₂ = π, where k = (k₁, k₂) is the total quasimomentum, has an infinite number of eigenvalues. In the case of a special potential, one eigenvalue is simple, another one is double, and the other eigenvalues have multiplicity three. We prove that the double eigenvalue of H(π,π) splits into two nondegenerate eigenvalues of H(π, π − 2β) for small β > 0 and the eigenvalues of multiplicity three similarly split into three different nondegenerate eigenvalues. We obtain asymptotic formulas with the accuracy of β² and also an explicit form of the eigenfunctions of H(π, π −2β) for these eigenvalues.