On the Existence of Eigenvalues of the Three-Particle Discrete Schr\"{o}dinger Operator

Zh. I. Abdullaev ; J. Kh. Boymurodov ; A. M. Khalkhuzhaev

Matematičeskie zametki, Tome 114 (2023) no. 5, pp. 643-658

Résumé

We consider the three-particle Schrödinger operator $H_{\mu,\lambda,\gamma} (\mathbf K)$, $\mathbf K\in \mathbb{T}^3$, associated with a system of three particles (of which two are bosons with mass $1$ and one is arbitrary with mass $m=1/\gamma1$) coupled by pairwise contact potentials $\mu>0$ and $\lambda>0$ on the three-dimensional lattice $\mathbb{Z}^3$. We prove that there exist critical mass ratio values $\gamma=\gamma_{1}$ and $\gamma=\gamma_{2}$ such that for sufficiently large $\mu>0$ and fixed $\lambda>0$ the operator $H_{\mu,\lambda,\gamma}(\mathbf{0})$, $\mathbf{0}=(0,0,0)$, has at least one eigenvalue lying to the left of the essential spectrum for $\gamma\in (0,\gamma_{1})$, at least two such eigenvalues for $\gamma\in (\gamma_{1},\gamma_{2})$, and at least four such eigenvalues for $\gamma\in (\gamma_{2}, +\infty)$.

Détail

Keywords: Schrödinger operator, lattice, Hamiltonian, zero-range potential, eigenvalue, total quasimomentum, invariant subspace, Faddeev operator.
Mots-clés : boson