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

We consider a Hamiltonian of a two-boson system on a two-dimensional lattice $\mathbb Z^2$. The Schrödinger operator $H(k_1,k_2)$ of the system for $k_1=k_2= \pi$, where $\mathbf k=(k_1,k_2)$ 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 $H(\pi,\pi)$ splits into two nondegenerate eigenvalues of $H(\pi,\pi-2\beta)$ for small $\beta>0$ and the eigenvalues of multiplicity three similarly split into three different nondegenerate eigenvalues. We obtain asymptotic formulas with the accuracy of $\beta^2$ and also an explicit form of the eigenfunctions of $H(\pi,\pi-2\beta)$ for these eigenvalues.