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

Zh. I. Abdullaev ; K. D. Kuliev

Teoretičeskaâ i matematičeskaâ fizika, Tome 186 (2016) no. 2, pp. 272-292

Résumé

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.

Détail

Keywords: Hamiltonian, bound state, Schrödinger operator, total quasimomentum, eigenvalue, perturbation theory, Birman–Schwinger principle.