On the Existence of Eigenvalues of the Three-Particle Discrete Schrödinger Operator
Abstract
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 $$m=1/\gamma<1$$ ) 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 $$\gamma\in (\gamma_{2}, +\infty)$$ .