Invariant Subspaces and Eigenvalues of the Three-Particle Discrete Schrödinger Operators

Abstract

We consider three-particle Schrödinger operator \({{H}_{{\mu ,\gamma }}}({\mathbf{K}})\), \({\mathbf{K}} \in {{\mathbb{T}}^{3}}\), associated to a system of three particles (two of them are bosons with mass 1 and one is an arbitrary with mass \(m = {\text{1/}}\gamma < 1\)), interacting via zero-range pairwise potentials \(\mu > 0\) and λ > 0 on the three dimensional lattice \({{\mathbb{Z}}^{3}}\). It is proved that there exist critical value of ratio of mass γ = γ₁ and γ = γ₂ such that the operator \({{H}_{{\mu ,\gamma }}}(\mathbf{0})\) 0 = (0, 0, 0), has a unique eigenvalue for \(\gamma \in (0,{{\gamma }_{1}})\), has two eigenvalues for \(\gamma \in ({{\gamma }_{1}},{{\gamma }_{2}})\) and four eigenvalues for \(\gamma \in ({{\gamma }_{2}}, + \infty )\), located on the left-hand side of the essential spectrum for large enough µ > 0 and fixed λ > 0.