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