On the Existence of Eigenvalues of a Three Particle System on a Cubic Lattice

We consider a three particle system on the lattice $$\mathbb{Z}^d$$ , $$d=1,2$$ , consisting of two identical bosons and one other particle interacting via pairwise attractive contact potentials. For a fixed $$K\in \mathbb{T}^d$$ , the system is described by the fiber Schrödinger operator $$H_{\lambda\mu}(K)$$ , where $$\lambda<0$$ and $$\mu<0$$ , respectively, characterize the interaction of two bosons and the interaction of a boson with another particle, and $$K$$ denotes the total quasi-momentum of the system. The existence of an eigenvalue of the operator $$H_{\lambda\mu}(K)$$ below the lower boundary of its essential spectrum is proved for any $$\lambda,\mu<0$$ . Furthermore, the analytical dependence of the eigenvalue and the corresponding eigenfunction on the parameter $$K\in \mathbb{T}^d$$ is established.